Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter example. 62. Every polynomial function is a rational function. 63. Every rational function is a polynomial function. 64. The graph of every rational function has at least one vertical asymptote. 65. The graph of every exponential function has a horizontal asymptote. 66. The graph of every logarithmic function has a vertical asymptote. 67. There exists a rational function that has both a vertical and horizontal asymptote.
The Correct Answer and Explanation is :
Let’s analyze each of these statements one by one, providing an explanation and counterexamples where applicable.
62. Every polynomial function is a rational function.
True. A polynomial function is any function of the form:
[
P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0
]
This is a sum of terms where the exponents of (x) are non-negative integers. A rational function is any function that can be expressed as a ratio of two polynomials, i.e.,
[
R(x) = \frac{P(x)}{Q(x)}
]
where (P(x)) and (Q(x)) are polynomials, and (Q(x) \neq 0). Every polynomial can be written as a rational function by expressing it as:
[
\frac{P(x)}{1}
]
Since 1 is a polynomial and non-zero, every polynomial is technically a rational function.
63. Every rational function is a polynomial function.
False. A rational function is a ratio of two polynomials. It is not necessarily a polynomial function. For example:
[
R(x) = \frac{1}{x}
]
This is a rational function but not a polynomial because it has a denominator. Polynomials do not have denominators with variables. Hence, not every rational function is a polynomial.
64. The graph of every rational function has at least one vertical asymptote.
False. Not all rational functions have vertical asymptotes. A vertical asymptote occurs when the denominator of a rational function approaches zero, but the numerator does not. For example:
[
R(x) = \frac{x+1}{x^2 + 1}
]
The denominator (x^2 + 1) never equals zero, so there is no vertical asymptote in this case. Thus, not every rational function has a vertical asymptote.
65. The graph of every exponential function has a horizontal asymptote.
True. Exponential functions generally take the form:
[
f(x) = a \cdot b^x
]
where (a) is a constant and (b) is a positive real number. As (x) approaches infinity or negative infinity, the function approaches a horizontal line, typically (y = 0). This means exponential functions have a horizontal asymptote, usually at (y = 0), unless the function is transformed (e.g., shifted vertically).
66. The graph of every logarithmic function has a vertical asymptote.
True. Logarithmic functions take the form:
[
f(x) = a \cdot \log_b(x – h) + k
]
For (x > h), the graph approaches negative infinity as (x) approaches (h), meaning there is a vertical asymptote at (x = h). This is true for all basic logarithmic functions, and the vertical asymptote occurs where the argument inside the logarithm is zero.
67. There exists a rational function that has both a vertical and horizontal asymptote.
True. A rational function can have both vertical and horizontal asymptotes. For example:
[
R(x) = \frac{1}{x^2 + 1}
]
This function has a vertical asymptote at (x = 0) (where the denominator approaches zero) and a horizontal asymptote at (y = 0) as (x) approaches infinity. Thus, such rational functions do exist.
Summary of Answers:
- 62: True (Every polynomial is a rational function).
- 63: False (Not every rational function is a polynomial).
- 64: False (Not every rational function has a vertical asymptote).
- 65: True (Every exponential function has a horizontal asymptote).
- 66: True (Every logarithmic function has a vertical asymptote).
- 67: True (There exists a rational function with both vertical and horizontal asymptotes).
I will generate a plot of rational functions with vertical and horizontal asymptotes for a visual understanding.
Here is a graph showing a rational function with both a vertical and horizontal asymptote, helping to visualize the behavior of such functions. The vertical asymptote occurs at (x = 0), and the horizontal asymptote is at (y = 0). This demonstrates how rational functions can have both types of asymptotes. Let me know if you need further explanations!