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 rational
functions are polynomials.
b. If f is a linear polynomial, then f ∘ f is a quadratic polynomial.
c. If f and g are polynomials, then the degrees of f ∘ g and g ∘ f
are equal.
d. To graph g(x) = f(x + 2), shift the graph of f 2 units to the
right.
The Correct Answer and Explanation is :
Let’s go through each of these statements one by one:
a. All polynomials are rational functions, but not all rational functions are polynomials.
- True.
- Explanation: A polynomial 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 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 ).
- 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.
b. If ( f ) is a linear polynomial, then ( f \sim f ) is a quadratic polynomial.
- False.
- Explanation: A linear polynomial has the form ( f(x) = ax + b ), where ( a ) and ( b ) are constants and ( a \neq 0 ). A quadratic polynomial 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.
c. If ( f ) and ( g ) are polynomials, then the degrees of ( f + g ) and ( g + f ) are equal.
- True.
- Explanation: 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.
d. To graph ( g(x) = f(x + 2) ), shift the graph of ( f ) 2 units to the right.
- False.
- Explanation: The transformation ( g(x) = f(x + 2) ) corresponds to a shift of the graph of ( f ) 2 units to the left, 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.
Summary:
- a. True – All polynomials are rational functions, but not all rational functions are polynomials.
- b. False – A linear polynomial does not result in a quadratic polynomial.
- c. True – The degrees of ( f + g ) and ( g + f ) are equal.
- d. False – The graph of ( g(x) = f(x + 2) ) is shifted 2 units to the left, not the right.
I’ll now generate an image to help visualize these concepts.
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.