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

The rational function $f\left(x\right)=\frac{2x^2+5x-3}{x^2-9}$ can be rewritten in the equivalent factored form $f\left(x\right)=\frac{\left(2x-1\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}$ . Which of the following features of the graph of $f$ is most readily revealed by this factored form?

A manufacturer determines that the total cost, in dollars, to produce $x$ units of a product is given by $C\left(x\right)=500+12x$. The average cost per unit is modeled by the rational function $A\left(x\right)=\frac{500+12x}{x}$. What is the value of $A\left(50\right)$ and what does it represent in this context?

The function $r$ is given by $r\left(x\right)=\frac{x^2-16}{x^2+7x+12}$.

At what value of $x$ does the graph of $r$ have a hole?

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

What are all real zeros of $r$?

In the $xy$-plane, the graph of a rational function $f$ has a vertical asymptote at $x=3$. Which of the following could be an expression for $f\left(x\right)$ ?

The rational function $r$ is given by $r\left(x\right)=\frac{4x^2-3x+1}{2x^2+5}$ . Which of the following describes the end behavior of $r$ as $x$ decreases without bound?

Which of the following functions has a zero at $x=-2$ and has a graph in the $xy$-plane with a vertical asymptote at $x=3$ and a hole at $x=-1$ ?

The function $g$ is given by $g\left(x\right)=x^3-x^2-6x$, and the function $h$ is given by $h\left(x\right)=x^2+4x-5$. Let $k$ be the function given by $k\left(x\right)=\frac{h\left(x\right)}{g\left(x\right)}$. What is the domain of $k$?

The rational function $g$ is given by $g\left(x\right)=\frac{x+2}{{\left(x-3\right)}^2}$ . Which of the following correctly describes the unbounded behavior of $g$ near $x=3$ ?

The graph of the rational function $h\left(x\right)=\frac{x^2+2x-8}{x-2}$ has a hole at $x=2$. What are the coordinates of the hole?

The function $r$ is given by $r\left(x\right)=\frac{x+5}{x^2-9x+18}$.

What is the domain of $r$?

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

The graph of $r$ in the $xy$-plane has one hole. Which of the following are the coordinates of the hole in the graph of $r$?

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

Which of the following is the equation of the slant asymptote of the graph of $p$?

Let $p\left(x\right)=\frac{x-4}{x+1}$ . On which of the following intervals is $p\left(x\right)\ge 0$ ?

Let $f$ be a rational function that is graphed in the $xy$-plane. Consider $x=2$ and $x=-4$. The polynomial in the numerator of $f$ has a zero at $x=2$ and does not have a zero at $x=-4$. The polynomial in the denominator of $f$ has zeros at both $x=2$ and $x=-4$. The multiplicities of the zeros at $x=2$ in the numerator and in the denominator are equal. Which of the following statements is true?

The function $m$ is given by $m\left(x\right)=\frac{2x^2-5x+1}{x-3}$ . The graph of $m$ has a slant asymptote. Which of the following is the equation of the slant asymptote?

Consider the function $f$ defined by $f\left(x\right)=\frac{x^2-4}{x^2-x-6}$. Which of the following correctly describes the features of the graph of $f$?

The graph of the function $f\left(x\right)=\frac{x^3-4x}{x^2-2x}$ has holes at the points $\left(0,a\right)$ and $\left(2,b\right)$. What is the value of $a+b$?