Precalculus Rational Functions and Asymptotes Worksheet (PDF)

Precalculus math worksheet: Rational Functions and Asymptotes – Maths PDF

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This rational functions worksheet for Precalculus covers rational functions and asymptotes (Common Core A-APR.D.7, F-IF.C.7d, A-CED.A.1), with a free PDF to print.

These 20 problems practice rational functions and their graphs (Common Core A-APR.D.7, F-IF.C.7d and A-CED.A.1). You will find domains, asymptotes, holes and intercepts, solve rational equations and inequalities and work with applications.

Precalculus · Common Core State Standards: A-APR.D.7, F-IF.C.7d, A-CED.A.1

Rational functions worksheet: exercises 1 to 10

Exercise 1 – Domain

Find the domain of $f(x)=\frac{x+1}{x^{2}-5x+6}$.

Exercise 2 – Hole and asymptote

Find the vertical asymptote and hole of $f(x)=\frac{x^{2}-9}{x^{2}-x-6}$.

Exercise 3 – Equal degrees

Find the horizontal asymptote of $f(x)=\frac{6x^{2}-5}{2x^{2}+x}$.

Exercise 4 – Lower numerator degree

Find the horizontal asymptote of $f(x)=\frac{2x+1}{x^{2}+3}$.

Exercise 5 – No horizontal asymptote

Does $f(x)=\frac{x^{4}+1}{x^{2}-1}$ have a horizontal or slant asymptote?

Exercise 6 – Intercepts

Find the intercepts of $f(x)=\frac{2x-6}{x+4}$.

Exercise 7 – Read a graph

Graph of a rational function.

The graph of $f(x)=\frac{2x}{x-3}$ is shown. State the equations of the asymptotes and the intercept.

Exercise 8 – Slant asymptote

Graph of f.

Find the slant asymptote of $f(x)=\frac{x^{2}-x-2}{x-3}$.

Exercise 9 – Find the hole

Graph with a missing point.

Find the coordinates of the hole in the graph of $f(x)=\frac{x^{2}-4}{x-2}$.

Exercise 10 – Solve a proportion

Solve $\frac{3}{x-1}=\frac{2}{x+2}$.

Exercises 11 and up

Exercise 11 – Solve with two fractions

Solve $\frac{2}{x}+\frac{3}{x+1}=2$.

Exercise 12 – Extraneous solution

Solve $\frac{x}{x-3}=\frac{3}{x-3}+2$.

Exercise 13 – Rational inequality

Critical values -2 and 1.

Solve $\frac{x-1}{x+2}\ge 0$.

Exercise 14 – Write an equation

Write a rational function with vertical asymptote $x=2$, horizontal asymptote $y=3$ and $x$-intercept $-1$.

Exercise 15 – Behavior near an asymptote

For $f(x)=\frac{1}{x-2}$, find $f(2.01)$ and $f(1.99)$, and describe the behavior as $x\to 2^{+}$ and $x\to 2^{-}$.

Exercise 16 – Hole and asymptote together

Simplify $f(x)=\frac{(x-1)(x+3)}{(x-1)(x-5)}$ and find the hole and vertical asymptote.

Exercise 17 – Average cost of shirts

Average cost per shirt.

Printing $n$ shirts for the school store costs $\$500$ plus $\$12$ per shirt, so the average cost per shirt is $A(n)=\frac{500+12n}{n}$. Find $A(50)$, identify the horizontal asymptote and explain it, and find the smallest whole $n$ for an average of at most $\$15$.

Exercise 18 – Salt mixture

Salt fraction after adding salt.

A tank holds $10$ gallons of a $20\%$ salt solution. Aisha adds $x$ gallons of pure salt. The salt fraction is $c(x)=\frac{2+x}{10+x}$. Find the horizontal asymptote and find $x$ for a $50\%$ solution.

Exercise 19 – Painting a fence

Jamal takes $3$ hours less than Mei to paint a fence alone. Together they take $2$ hours. Let $x$ be Mei’s time. Solve $\frac1x+\frac1{x-3}=\frac12$ and find both times.

Exercise 20 – Complete analysis

Graph of f.

For $f(x)=\frac{x^{2}-4}{x^{2}-x-6}$, find the hole, vertical asymptote, horizontal asymptote and both kinds of intercept.

Key facts behind this rational functions worksheet

Key facts

Domain: all real numbers except zeros of the denominator.
Vertical asymptote: a denominator zero that does not cancel. Hole: a cancelled factor.
Degree of numerator < denominator: y = 0. Equal degrees: y = ratio of leading coefficients.
Degree of numerator = degree of denominator + 1: slant asymptote from division.
Always check rational equation answers for extraneous solutions.

How to use this rational functions worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the rational functions worksheet as a PDF with the button below and keep it with your notes.

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