Domain and Range of a Rational Function

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

For a rational function, the original denominator determines forbidden inputs. Factoring can simplify the formula but does not put those inputs back into the domain. The range needs a separate argument.

Method

To find the range, solve $y=p(x)/q(x)$ for $x$ , then check for impossible targets and forbidden solutions. Removable holes can exclude an otherwise attainable output.

Worked example

For $f(x)=(x+1)/(x-2)$ , the domain excludes $2$ . Solve $y(x-2)=x+1$ :

x=\frac{2y+1}{y-1},

y\ne1.

This expression never equals $2$ , so every $y\ne1$ is attained. At $y=1$ , the equation would say $-2=1$ , impossible. The range excludes $1$ .

Common mistake

For $(x^2-1)/(x-1)$ , canceling gives $x+1$ only when $x\ne1$ . The hole means the range also excludes $2$ .

Check your understanding

Find the domain and range of $g(x)=1/(x+3)+2$ .

Show answer

Domain: all real $x$ except $-3$ . Range: all real $y$ except $2$ .

Explore

Sign of a rational expression

Try this. Move the test point across −2 and 1. A zero may be included in a non-strict inequality; a denominator zero never can.

Test x0.5

(x − 1)/(x + 2) at x = 0.5: -0.2 · negative. Positive on (−∞, −2) ∪ (1, ∞); x = −2 is always excluded.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

What is the domain of f(x)=1/(x−3)?

x > 3

All real x except 3

All real x except 0

Hint 1 · Find a starting point

A denominator cannot be zero.

Hint 2 · Take the next step

Solve x−3=0 to locate the excluded input.

Show the reasoning

Answer: All real x except 3

Only x=3 makes the expression undefined; negative denominators are allowed.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

Optional marks, not a grade. Saved in this browser only. Open notebook →