Let’s go through each of these statements one by one:<\/p>\n\n\n\n
a. All polynomials are rational functions, but not all rational functions are polynomials.<\/h3>\n\n\n\n\n- True.<\/strong><\/li>\n\n\n\n
- Explanation:<\/strong> A polynomial<\/strong> is a function of the form ( f(x) = a_nx^n + a_{n-1}x^{n-1} + \\dots + a_1x + a_0 ), where ( a_n, a_{n-1}, \\dots, a_0 ) are constants and ( n ) is a non-negative integer. A rational function<\/strong> is defined as a quotient of two polynomials, i.e., ( g(x) = \\frac{p(x)}{q(x)} ), where ( p(x) ) and ( q(x) ) are polynomials, and ( q(x) \\neq 0 ).<\/li>\n\n\n\n
- Since a polynomial can be written as ( \\frac{p(x)}{1} ), it is a rational function. However, not all rational functions are polynomials. For example, ( g(x) = \\frac{1}{x} ) is a rational function but not a polynomial.<\/li>\n<\/ul>\n\n\n\n
b. If ( f ) is a linear polynomial, then ( f \\sim f ) is a quadratic polynomial.<\/h3>\n\n\n\n\n- False.<\/strong><\/li>\n\n\n\n
- Explanation:<\/strong> A linear polynomial<\/strong> has the form ( f(x) = ax + b ), where ( a ) and ( b ) are constants and ( a \\neq 0 ). A quadratic polynomial<\/strong> has the form ( f(x) = ax^2 + bx + c ), where ( a \\neq 0 ). The statement ( f \\sim f ) does not make sense in this context. If the question refers to the operation ( f(x) + f(x) ), the result will still be a linear polynomial because the highest degree of ( f(x) ) is 1, so adding it to itself will still not produce a quadratic term. Thus, a linear polynomial added to itself does not result in a quadratic polynomial.<\/li>\n<\/ul>\n\n\n\n
c. If ( f ) and ( g ) are polynomials, then the degrees of ( f + g ) and ( g + f ) are equal.<\/h3>\n\n\n\n\n- True.<\/strong><\/li>\n\n\n\n
- Explanation:<\/strong> The degree of a polynomial is the highest exponent of ( x ) in the polynomial. Addition of polynomials is commutative, meaning ( f + g = g + f ). The degree of the sum of two polynomials depends on the highest degree of ( f ) or ( g ). Therefore, the degrees of ( f + g ) and ( g + f ) are always equal.<\/li>\n<\/ul>\n\n\n\n
d. To graph ( g(x) = f(x + 2) ), shift the graph of ( f ) 2 units to the right.<\/h3>\n\n\n\n\n- False.<\/strong><\/li>\n\n\n\n
- Explanation:<\/strong> The transformation ( g(x) = f(x + 2) ) corresponds to a shift<\/strong> of the graph of ( f ) 2 units to the left<\/strong>, not to the right. In general, if a function is of the form ( g(x) = f(x + h) ), then the graph of ( f(x) ) is shifted ( h ) units horizontally. If ( h ) is positive, the shift is to the left; if ( h ) is negative, the shift is to the right.<\/li>\n<\/ul>\n\n\n\n
Summary:<\/h3>\n\n\n\n\n- a.<\/strong> True \u2013 All polynomials are rational functions, but not all rational functions are polynomials.<\/li>\n\n\n\n
- b.<\/strong> False \u2013 A linear polynomial does not result in a quadratic polynomial.<\/li>\n\n\n\n
- c.<\/strong> True \u2013 The degrees of ( f + g ) and ( g + f ) are equal.<\/li>\n\n\n\n
- d.<\/strong> False \u2013 The graph of ( g(x) = f(x + 2) ) is shifted 2 units to the left, not the right.<\/li>\n<\/ul>\n\n\n\n
I’ll now generate an image to help visualize these concepts.<\/p>\n\n\n\n![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/03\/image-5.png\")
<\/figure>\n\n\n\nHere is a visual representation of the transformation of the function ( f(x) ) to ( g(x) = f(x + 2) ), showing the graph shifted 2 units to the left. This illustrates the correct interpretation of the horizontal shift.<\/p>\n","protected":false},"excerpt":{"rendered":"
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. All polynomials are rational functions, but not all rationalfunctions are polynomials.b. If f is a linear polynomial, then f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 f is a quadratic polynomial.c. If f and g are polynomials, then the degrees of f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-195838","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=195838"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
b. If ( f ) is a linear polynomial, then ( f \\sim f ) is a quadratic polynomial.<\/h3>\n\n\n\n\n- False.<\/strong><\/li>\n\n\n\n
- Explanation:<\/strong> A linear polynomial<\/strong> has the form ( f(x) = ax + b ), where ( a ) and ( b ) are constants and ( a \\neq 0 ). A quadratic polynomial<\/strong> has the form ( f(x) = ax^2 + bx + c ), where ( a \\neq 0 ). The statement ( f \\sim f ) does not make sense in this context. If the question refers to the operation ( f(x) + f(x) ), the result will still be a linear polynomial because the highest degree of ( f(x) ) is 1, so adding it to itself will still not produce a quadratic term. Thus, a linear polynomial added to itself does not result in a quadratic polynomial.<\/li>\n<\/ul>\n\n\n\n
c. If ( f ) and ( g ) are polynomials, then the degrees of ( f + g ) and ( g + f ) are equal.<\/h3>\n\n\n\n\n- True.<\/strong><\/li>\n\n\n\n
- Explanation:<\/strong> The degree of a polynomial is the highest exponent of ( x ) in the polynomial. Addition of polynomials is commutative, meaning ( f + g = g + f ). The degree of the sum of two polynomials depends on the highest degree of ( f ) or ( g ). Therefore, the degrees of ( f + g ) and ( g + f ) are always equal.<\/li>\n<\/ul>\n\n\n\n
d. To graph ( g(x) = f(x + 2) ), shift the graph of ( f ) 2 units to the right.<\/h3>\n\n\n\n\n- False.<\/strong><\/li>\n\n\n\n
- Explanation:<\/strong> The transformation ( g(x) = f(x + 2) ) corresponds to a shift<\/strong> of the graph of ( f ) 2 units to the left<\/strong>, not to the right. In general, if a function is of the form ( g(x) = f(x + h) ), then the graph of ( f(x) ) is shifted ( h ) units horizontally. If ( h ) is positive, the shift is to the left; if ( h ) is negative, the shift is to the right.<\/li>\n<\/ul>\n\n\n\n
Summary:<\/h3>\n\n\n\n\n- a.<\/strong> True \u2013 All polynomials are rational functions, but not all rational functions are polynomials.<\/li>\n\n\n\n
- b.<\/strong> False \u2013 A linear polynomial does not result in a quadratic polynomial.<\/li>\n\n\n\n
- c.<\/strong> True \u2013 The degrees of ( f + g ) and ( g + f ) are equal.<\/li>\n\n\n\n
- d.<\/strong> False \u2013 The graph of ( g(x) = f(x + 2) ) is shifted 2 units to the left, not the right.<\/li>\n<\/ul>\n\n\n\n
I’ll now generate an image to help visualize these concepts.<\/p>\n\n\n\n<\/figure>\n\n\n\nHere is a visual representation of the transformation of the function ( f(x) ) to ( g(x) = f(x + 2) ), showing the graph shifted 2 units to the left. This illustrates the correct interpretation of the horizontal shift.<\/p>\n","protected":false},"excerpt":{"rendered":"
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. All polynomials are rational functions, but not all rationalfunctions are polynomials.b. If f is a linear polynomial, then f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 f is a quadratic polynomial.c. If f and g are polynomials, then the degrees of f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-195838","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=195838"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
c. If ( f ) and ( g ) are polynomials, then the degrees of ( f + g ) and ( g + f ) are equal.<\/h3>\n\n\n\n\n- True.<\/strong><\/li>\n\n\n\n
- Explanation:<\/strong> The degree of a polynomial is the highest exponent of ( x ) in the polynomial. Addition of polynomials is commutative, meaning ( f + g = g + f ). The degree of the sum of two polynomials depends on the highest degree of ( f ) or ( g ). Therefore, the degrees of ( f + g ) and ( g + f ) are always equal.<\/li>\n<\/ul>\n\n\n\n
d. To graph ( g(x) = f(x + 2) ), shift the graph of ( f ) 2 units to the right.<\/h3>\n\n\n\n\n- False.<\/strong><\/li>\n\n\n\n
- Explanation:<\/strong> The transformation ( g(x) = f(x + 2) ) corresponds to a shift<\/strong> of the graph of ( f ) 2 units to the left<\/strong>, not to the right. In general, if a function is of the form ( g(x) = f(x + h) ), then the graph of ( f(x) ) is shifted ( h ) units horizontally. If ( h ) is positive, the shift is to the left; if ( h ) is negative, the shift is to the right.<\/li>\n<\/ul>\n\n\n\n
Summary:<\/h3>\n\n\n\n\n- a.<\/strong> True \u2013 All polynomials are rational functions, but not all rational functions are polynomials.<\/li>\n\n\n\n
- b.<\/strong> False \u2013 A linear polynomial does not result in a quadratic polynomial.<\/li>\n\n\n\n
- c.<\/strong> True \u2013 The degrees of ( f + g ) and ( g + f ) are equal.<\/li>\n\n\n\n
- d.<\/strong> False \u2013 The graph of ( g(x) = f(x + 2) ) is shifted 2 units to the left, not the right.<\/li>\n<\/ul>\n\n\n\n
I’ll now generate an image to help visualize these concepts.<\/p>\n\n\n\n<\/figure>\n\n\n\nHere is a visual representation of the transformation of the function ( f(x) ) to ( g(x) = f(x + 2) ), showing the graph shifted 2 units to the left. This illustrates the correct interpretation of the horizontal shift.<\/p>\n","protected":false},"excerpt":{"rendered":"
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. All polynomials are rational functions, but not all rationalfunctions are polynomials.b. If f is a linear polynomial, then f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 f is a quadratic polynomial.c. If f and g are polynomials, then the degrees of f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-195838","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=195838"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
d. To graph ( g(x) = f(x + 2) ), shift the graph of ( f ) 2 units to the right.<\/h3>\n\n\n\n\n- False.<\/strong><\/li>\n\n\n\n
- Explanation:<\/strong> The transformation ( g(x) = f(x + 2) ) corresponds to a shift<\/strong> of the graph of ( f ) 2 units to the left<\/strong>, not to the right. In general, if a function is of the form ( g(x) = f(x + h) ), then the graph of ( f(x) ) is shifted ( h ) units horizontally. If ( h ) is positive, the shift is to the left; if ( h ) is negative, the shift is to the right.<\/li>\n<\/ul>\n\n\n\n
Summary:<\/h3>\n\n\n\n\n- a.<\/strong> True \u2013 All polynomials are rational functions, but not all rational functions are polynomials.<\/li>\n\n\n\n
- b.<\/strong> False \u2013 A linear polynomial does not result in a quadratic polynomial.<\/li>\n\n\n\n
- c.<\/strong> True \u2013 The degrees of ( f + g ) and ( g + f ) are equal.<\/li>\n\n\n\n
- d.<\/strong> False \u2013 The graph of ( g(x) = f(x + 2) ) is shifted 2 units to the left, not the right.<\/li>\n<\/ul>\n\n\n\n
I’ll now generate an image to help visualize these concepts.<\/p>\n\n\n\n<\/figure>\n\n\n\nHere is a visual representation of the transformation of the function ( f(x) ) to ( g(x) = f(x + 2) ), showing the graph shifted 2 units to the left. This illustrates the correct interpretation of the horizontal shift.<\/p>\n","protected":false},"excerpt":{"rendered":"
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. All polynomials are rational functions, but not all rationalfunctions are polynomials.b. If f is a linear polynomial, then f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 f is a quadratic polynomial.c. If f and g are polynomials, then the degrees of f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-195838","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=195838"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
Summary:<\/h3>\n\n\n\n\n- a.<\/strong> True \u2013 All polynomials are rational functions, but not all rational functions are polynomials.<\/li>\n\n\n\n
- b.<\/strong> False \u2013 A linear polynomial does not result in a quadratic polynomial.<\/li>\n\n\n\n
- c.<\/strong> True \u2013 The degrees of ( f + g ) and ( g + f ) are equal.<\/li>\n\n\n\n
- d.<\/strong> False \u2013 The graph of ( g(x) = f(x + 2) ) is shifted 2 units to the left, not the right.<\/li>\n<\/ul>\n\n\n\n
I’ll now generate an image to help visualize these concepts.<\/p>\n\n\n\n<\/figure>\n\n\n\nHere is a visual representation of the transformation of the function ( f(x) ) to ( g(x) = f(x + 2) ), showing the graph shifted 2 units to the left. This illustrates the correct interpretation of the horizontal shift.<\/p>\n","protected":false},"excerpt":{"rendered":"
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. All polynomials are rational functions, but not all rationalfunctions are polynomials.b. If f is a linear polynomial, then f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 f is a quadratic polynomial.c. If f and g are polynomials, then the degrees of f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-195838","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=195838"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
I’ll now generate an image to help visualize these concepts.<\/p>\n\n\n\n Here is a visual representation of the transformation of the function ( f(x) ) to ( g(x) = f(x + 2) ), showing the graph shifted 2 units to the left. This illustrates the correct interpretation of the horizontal shift.<\/p>\n","protected":false},"excerpt":{"rendered":" Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.a. All polynomials are rational functions, but not all rationalfunctions are polynomials.b. If f is a linear polynomial, then f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 f is a quadratic polynomial.c. If f and g are polynomials, then the degrees of f \u00c3\u00a2\u00cb\u2020\u00cb\u0153 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-195838","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=195838"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195838\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195838"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195838"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195838"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}<\/figure>\n\n\n\n