sagnuhh
2021-06-27
Answered

Give the range of the rational functions given that neither of the graphs crosses its horizontal asymptote. Write your answers in set notation and interval notation.

You can still ask an expert for help

Nathanael Webber

Answered 2021-06-28
Author has **117** answers

1. For a rational function, the range is all real numbers, except the y values that correspond to holes or horizontal asymptotes.

In Example B, part(b), the horizontal asymptote was

The set notation for the range is

The interval notation is

2. In ‘Try These B, part(a), we were given

In a rational function, if the degree of the numerator and the denominator are the same, then the horizontal asymptote is given by

In this case, we have

Therefore, the set notation for the range is

The interval notation is

asked 2021-02-25

True or False. The graph of a rational function may intersect a horizontal asymptote.

asked 2022-04-10

Here is the problem.

A nonconstant rational function over the real numbers(a rational function is a function that can be expressed as $\frac{p(x)}{q(x)}$, with $p(x)$,$q(x)$ as polynomial function) $f(x)$, is defined such that:

$(f(x){)}^{2}-a=f({x}^{2})$ for all $x$ ( $a$ is a constant value)

Prove that $f(x)$ must be of the form ${x}^{k}$ for some constant $k$.

I have an idea of how to prove it, but I can't do so rigorously. I can prove that any polynomial with more than 2 terms, or monomials, would not work. But I can't prove it won't work for any rational function.

A nonconstant rational function over the real numbers(a rational function is a function that can be expressed as $\frac{p(x)}{q(x)}$, with $p(x)$,$q(x)$ as polynomial function) $f(x)$, is defined such that:

$(f(x){)}^{2}-a=f({x}^{2})$ for all $x$ ( $a$ is a constant value)

Prove that $f(x)$ must be of the form ${x}^{k}$ for some constant $k$.

I have an idea of how to prove it, but I can't do so rigorously. I can prove that any polynomial with more than 2 terms, or monomials, would not work. But I can't prove it won't work for any rational function.

asked 2021-06-21

True or False. Some rational functions have holes in their graph.

asked 2022-02-18

A rational function is a function f of the form $\frac{p}{q}$ where p and q are polynomial functions. The domain of f is $\{x\in \mathbb{R}:q\left(x\right)\ne 0\}$ . Prove every rational function is continuous.

I have previously proved that every polynomial function$p\left(x\right)={a}_{0}+{a}_{1}x+\dots +\dots {a}_{n}{x}^{n}$ is continuous on $\mathbb{R}$ . Could I use a theorem that states $\frac{f}{g}$ is continuous at $x}_{0$ if $g\left({x}_{0}\right)\ne 0$ ? Then $p\left(x\right)$ (which is already proved to be continuous) over $q\left(x\right)$ would be continuous?

I have previously proved that every polynomial function

asked 2021-05-17

Using the following table of values for the rational functions f, s, and h.

Based rule that defines the functions f, g, and h. How does the degree of the numerator compare to the degree of the denominator, (e.g., greater than, less than, or equal to.)?

asked 2021-06-13

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x: Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits.

asked 2021-06-23

Graph the rational functions. $y=\frac{{x}^{2}}{x+1}$