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Rational Functions (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 9 April 2026

Rational functions

What is a rational function?

A rational function is a function that can be written as a quotient of two polynomial functions
- $r (x) = \frac{p (x)}{q (x)}$
  - where $p (x)$ is the polynomial in the numerator
  - and $q (x)$ is the polynomial in the denominator
A rational function gives a measure of the relative size of the numerator polynomial compared to the denominator polynomial
- for each value of $x$ in the function's domain

What is the domain of a rational function?

A rational function is undefined wherever the denominator equals zero
- because division by zero is not defined
Unless otherwise specified in a question, the domain of a rational function is
- all real numbers $x$ for which the denominator is not zero
To find the domain, set the denominator equal to zero and solve
- the domain is all real numbers except those values
- E.g. for $r (x) = \frac{x^{2} + 3}{x - 1}$
  - The denominator is zero when $x - 1 = 0$ , i.e. when $x = 1$
  - The domain is all real numbers where $x \neq 1$
- Or for $r (x) = \frac{x + 5}{x^{2} - 9}$
  - The denominator is zero when $x^{2} - 9 = 0$
    - i.e. $(x - 3) (x + 3) = 0$ , so $x = 3$ or $x = - 3$
  - The domain is all real numbers where $x \neq 3$ and $x \neq - 3$

Examiner Tips and Tricks

Values excluded from the domain may correspond to vertical asymptotes or holes in the graph, but either way they are excluded from the domain.

How can I evaluate or solve equations involving rational functions?

To evaluate a rational function at a specific input, substitute the value into both the numerator and denominator and compute the quotient
- E.g. if $r (x) = \frac{x^{2} + 3}{x - 1}$
  - then $r (4) = \frac{16 + 3}{4 - 1} = \frac{19}{3}$
To solve an equation like $r (x) = k$ for a specific value $k$
- If the function is simple enough, you can solve algebraically
- For more complex rational functions, use a graphing calculator
  - Graph $y = r (x)$ and $y = k$ on the same set of axes, then find the intersection point(s)
    - The $x$ coordinates of the intersections will be the solutions to the equation
  - You may also be able to use your calculator's equation solving features to solve the equation directly
  - Report answers as decimal approximations accurate to three decimal places

Worked Example

The function $g$ is given by $g (x) = \frac{x^{3} - 7 x - 43}{3 - x}$ .

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

Answer:

The phrase "as decimal approximations" lets you know that this is meant to be solved using your calculator, rather than attempting to do it by hand

Graph the function on your graphing calculator and graph the horizontal line $y = - 4$ on the same set of axes

Then use the solving feature to find the coordinates of the point where the two graphs intersect

Graph showing a curve intersecting the horizontal line y=-4 at point labelled (4.27264, -4), with numbered coordinate axes.

So $g (x) = - 4$ when $x = 4.27264$

Round to 3 decimal places for your final answer

$x = 4.273$

Examiner Tips and Tricks

When using your graphing calculator to solve an equation like the one in the Worked Example

be sure to zoom out sufficiently far to make sure you haven't missed any additional points of intersection

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Rational Functions (College Board AP® Precalculus): Revision Note

Rational functions

What is a rational function?

What is the domain of a rational function?

Examiner Tips and Tricks

How can I evaluate or solve equations involving rational functions?

Worked Example

Examiner Tips and Tricks

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis