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Grade 12 · MHF4U · Advanced Functions

MHF4U Rational Functions Practice Test (with Solutions)

Rational Functions is where a polynomial lands in the denominator and the graph picks up asymptotes and holes. Expect to find the domain and range, the vertical and horizontal asymptotes, holes, zeros and intercepts of a rational function, and the unit goes on to rational equations and inequalities. Try each question below on paper, then open the answer.

Free Rational Functions Practice Test

8 questions, exactly as a free account gets them in the first Rational Functions test on Martian Lab. Try each one on paper first, then open the answer.

  1. What is the domain of the function f(x)=125x−1f(x) = \dfrac{12}{5x-1}?
    Show answer
    {x∈R, x≠15}\left\{x \in R,\ x \neq \dfrac{1}{5}\right\}
  2. What is the horizontal asymptote of f(x)=6−2x6x−3f(x) = \dfrac{6-2x}{6x-3}?
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    y=−13y = \dfrac{-1}{3}
  3. What are the zeros of f(x)=(x2−4)(x−1)(x−2)(x+3)f(x) = \dfrac{(x^2-4)(x-1)}{(x-2)(x+3)}?
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    x=1,−2x = 1, -2
  4. What is the y-intercept of f(x)=(x−3)(2x+1)x+4f(x) = \dfrac{(x-3)(2x+1)}{x+4}?
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    y=−34y = \dfrac{-3}{4}
  5. What is the equation of the vertical asymptote of f(x)=(x−2)(x−1)2(x−1)(x+3)f(x) = \dfrac{(x-2)(x-1)^2}{(x-1)(x+3)}?
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    x=−3x = -3
  6. What are the coordinates of the hole (pinhole) of f(x)=x2−x−2x2+4x+3f(x) = \dfrac{x^2-x-2}{x^2+4x+3}?
    Show answer
    (x−2)(x+1)(x+3)(x+1)\dfrac{(x-2)(x+1)}{(x+3)(x+1)} →\rightarrow (−1, −32)\left(-1,\ \dfrac{-3}{2}\right)
  7. What is the domain of f(x)=−1x2+5f(x) = -\dfrac{1}{x^2+5}?
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    {x∈R}\{x \in R\}
  8. What is the range of f(x)=−1x2+5f(x) = -\dfrac{1}{x^2+5}?
    Show answer
    {y∈R, −15≤y<0}\left\{y \in R,\ \dfrac{-1}{5} \leq y < 0\right\}

This Unit on Martian Lab

The lessons start with reciprocal functions and their graphs, move to the intercepts and asymptotes of rational functions, and finish with rational inequalities, equations and applications.

  1. Reciprocal Functions
  2. Graphing Reciprocal Functions
  3. Intercepts & Asymptotes
  4. Intercepts & Asymptotes of Rational Functions
  5. Rational Inequalities
  6. Rational Equations and Applications

Then 4 full Rational Functions practice tests to check yourself.

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Rational Functions FAQ

What is on an MHF4U rational functions test?+

Reading a rational function: its domain and range, vertical and horizontal asymptotes, holes, zeros and intercepts. The unit also covers rational equations and inequalities. The free questions on this page focus on reading the function.

How do you find the vertical asymptotes?+

Factor the top and bottom and cancel any common factor. Each zero left in the denominator gives a vertical asymptote x = a. A factor that cancelled gives a hole instead.

How do you find the horizontal asymptote?+

Compare the degrees. If the bottom's degree is higher, it is y = 0. If the degrees match, it is the ratio of the leading coefficients. If the top's degree is higher, there is no horizontal asymptote.

What is a hole in a rational function?+

A point where a factor cancels from the top and bottom. The graph follows the simplified function but skips that one x value. Put the x value into the simplified function to get the hole's y coordinate.

Rational Functions Topics

One idea at a time: a short explanation, then five worked examples.

More MHF4U Units

Previous unit: Polynomial Equations

Next unit: Trigonometric Ratios

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