A function is one-to-one when no two inputs share an output. Only these functions have inverses.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The horizontal line test
Draw any horizontal line across the graph. If it ever meets the curve twice, the function is not one-to-one.
Why it decides
A horizontal line at height k marks every input giving output k. Two hits means the inverse would need two outputs, which no function may have.
Restricting the domain
y = x² fails the test, but on x ≥ 0 it passes. Cutting the domain is how the square root, arcsine and arccosine all come to exist.
The graph of an inverse
Swapping inputs and outputs swaps coordinates, so the graph of f⁻¹ is the graph of f reflected across y = x.
Domains trade places
The domain of f becomes the range of f⁻¹, and the range of f becomes the domain of f⁻¹.
Only some functions can be undone
A function has an inverse only if it is one-to-one: no two inputs share an output. Otherwise the inverse would have to send one input to two places, which no function can do.
The horizontal line test
If any horizontal line meets the graph more than once, two inputs share an output and the function is not invertible. It is the exact mirror of the vertical line test that decides whether something is a function at all.
Restricting the domain rescues it
x² is not one-to-one, but restricted to x ≥ 0 it is, and that restricted version has the square root as its inverse. The same move gives the inverse trigonometric functions their principal ranges.
Inverse graphs reflect in y = x
Swapping input and output swaps the coordinates of every point, which is a reflection in the line y = x. Domain and range exchange roles, so the inverse's domain is the original's range.
Step 2: Try It Yourself
Tap and try it out.
The slope is 1: for every 1 across, the line goes 1 up.
Step 3: Watch an Example
One step at a time.
Watch Rosa Restrict a Domain
Rosa needs an inverse for f(x) = x² − 4.
- Step 1
She checks the horizontal line test first, and the parabola fails it: 3 and −3 both give 5.
Step 4: Your Turn
Practice makes it stick.
The Point
Problem 1 of 2
The point (3, 8) lies on f. What is the x-coordinate of the matching point on f⁻¹?
The Test
Problem 2 of 2
Which function fails the horizontal line test?
Undo It
1 of 8
f(x) = 2x + 3. What is f⁻¹(9)?
2 of 8
(5, 2) lies on f⁻¹. What is f(2)?
3 of 8
f(x) = x³. What is f⁻¹(27)?
4 of 8
The range of f is y ≥ 4. What is the smallest value in the domain of f⁻¹?
5 of 8
f(x) = x² on x ≥ 0. What is f⁻¹(49)?
6 of 8
Which line does the graph of an inverse reflect across? Enter 1 for y = x, or 2 for the x-axis.
7 of 8
Which functions are one-to-one over all real numbers?
8 of 8
f(x) = (x − 1) ÷ 4. What is f⁻¹(2)?
Step 5: Quick Check
Show what you know.
Question 1 of 2
(2, 11) lies on f. What is f⁻¹(11)?
Question 2 of 2
Why must a function be one-to-one to have an inverse?
What You Learned
- A function has an inverse exactly when it is one-to-one.
- The horizontal line test decides it from the graph.
- Restricting the domain can rescue a function that fails.