Rational Functions (College Board AP® Precalculus): Exam Questions

The function $f$ is given by $f\left(x\right)=\frac{x^2+x-6}{x^2-9}$.

Find all real zeros of $f$.

Find the $x$-coordinate of any holes in the graph of $f$.

Find the equation of any vertical asymptote(s) of the graph of $f$.

The function $r$ is given by $r\left(x\right)=\frac{x^2-11}{x+4}$.

Find all values of $x$, as decimal approximations, for which $r\left(x\right)=0$, or indicate that there are no such values.

The function $g$ is given by $g\left(x\right)=\frac{2x^3-5x+1}{x-3}$.

(i) Find all values of $x$, as decimal approximations, for which $g\left(x\right)=-4$, or indicate that there are no such values.

(ii) Determine the end behavior of $g$ as $x$ increases without bound. Express your answer using the mathematical notation of a limit.

The rational function $f$ is given by $f\left(x\right)=\frac{x^2-3x-10}{x^2-25}$ .

Find all real zeros of $f\left(x\right)$. Show the work that leads to your answer.

The graph of $f$ has a hole at $x=5$. Determine the $y$-coordinate of the hole. Express your answer using the mathematical notation of a limit.

State the equation of the vertical asymptote of the graph of $f$. Express the unbounded behavior of $f$ near this asymptote using a one-sided limit as $x\to -5^{+}$.

The rational function $h$ is given by $h\left(x\right)=\frac{2x-8}{x^2+2x-3}$ .

Determine the domain of $h$.

Solve the inequality $h\left(x\right)\ge 0$. Show the work that leads to your answer, including the use of a sign test.

A bus rental company charges a fixed fee of $250 plus an additional $8 per passenger. If $x$ passengers ride the bus, the average cost per passenger, in dollars, is given by the rational function

$C\left(x\right)=\frac{250+8x}{x}$

for $x>0$.

Find $C\left(35\right)$, expressed as a decimal approximation. Include units in your answer.

Find $\underset{x\to \infty }{\lim }C\left(x\right)$ and interpret your answer in the context of the problem.

The function $r$ is given by

$r\left(x\right)=\frac{x^3-x^2-4x+4}{x-1}$

Use polynomial long division to rewrite $r\left(x\right)$ as a polynomial expression for all $x\ne 1$. Show the work that leads to your answer.

The graph of $r$ has a hole at $x=1$. Find the $y$-coordinate of the hole. Express your answer using the mathematical notation of a limit.

The function $g$ is given by $g\left(x\right)=\frac{3x^2-8x+5}{2x^2+7}$.

(i) Find all values of $x$, as decimal approximations, for which $g\left(x\right)=1$, or indicate that there are no such values.

(ii) Determine the end behavior of $g$ as $x$ increases without bound. Express your answer using the mathematical notation of a limit.

The rational function $g$ is given by $g\left(x\right)=\frac{x^3-2x^2+4x-5}{x^2+1}$ .

Use polynomial long division to rewrite $g\left(x\right)$ in the equivalent form $q\left(x\right)+\frac{r\left(x\right)}{x^2+1}$, where $q\left(x\right)$ and $r\left(x\right)$ are polynomials and the degree of $r\left(x\right)$ is less than 2. Show your work.

Determine the equation of the slant asymptote of the graph of $g$. Justify your answer using the result from part (a) and the end behavior of the remainder term as $x$ increases without bound.

A rational function $p$ is given in two equivalent analytic forms.

Standard form: $p\left(x\right)=\frac{3x^2-12x}{x^2-2x-8}$

Factored form: $p\left(x\right)=\frac{3x\left(x-4\right)}{\left(x-4\right)\left(x+2\right)}$

Determine the end behavior of $p$ as $x$ increases without bound. Express your answer using the mathematical notation of a limit.

Identify the $x$-coordinate of any holes in the graph of $p$.

Solve the equation $p\left(x\right)=2$, or indicate that there are no such values. Show the work that leads to your answer.

Let $r\left(x\right)=\frac{a{x}^2+bx+c}{x^2+dx-6}$, where $a$, $b$, $c$ and $d$ are constants.

The graph of $r$ has all of the following features:

• a horizontal asymptote at $y=2$

• a vertical asymptote at $x=3$

• a hole at $x=-2$

• an $x$-intercept at $x=1$

Use the horizontal asymptote to find the value of $a$.

Use the vertical asymptote and the hole to find the value of $d$. Show the work that leads to your answer.

Use the $x$-intercept and the hole to find the values of $b$ and $c$. Show the work that leads to your answer.