An inverse sends every output back to the input it came from. It exists only when no two inputs share an output.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The horizontal line test
If any horizontal line meets the graph twice, the function is not one-to-one and has no inverse.
Finding one
Swap x and y, then solve for y. The swap is what reverses the roles of input and output.
Verifying
Compose both ways. Both f(f⁻¹(x)) and f⁻¹(f(x)) must return x.
Restricting a domain
y = x² fails the test, but on x ≥ 0 it passes. Restricting is how the square root comes to exist at all.
The graph
Swapping coordinates reflects the graph across y = x, so an inverse is always that mirror image.
Only one-to-one functions can be undone
If two inputs share an output, an inverse would have to send one input to two places. The horizontal line test decides it, and restricting the domain can rescue functions that fail.
Swap and solve
Exchange x and y and solve for y. The swap is the statement that inputs and outputs have exchanged roles, which is the definition rather than a technique.
Reflection in y = x
The inverse graph is the original reflected in the line y = x, with domain and range exchanged. A point (a, b) on f corresponds to (b, a) on the inverse.
The standard inverse pairs
Squaring and square root on a restricted domain, exponential and logarithm, and the trigonometric functions with their principal ranges. Recognising the pairing is what makes equations solvable.
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 Sana Restrict and Invert
Sana needs an inverse for f(x) = x² − 4.
- Step 1
The parabola fails the horizontal line test, since 3 and −3 both give 5.
Step 4: Your Turn
Practice makes it stick.
The Inverse
Problem 1 of 2
f(x) = 3x + 6. What is f⁻¹(12)?
The Point
Problem 2 of 2
The point (3, 8) lies on f. What is f⁻¹(8)?
Undo It
1 of 8
f(x) = x + 5. What is f⁻¹(12)?
2 of 8
f(x) = 4x. What is f⁻¹(20)?
3 of 8
f(x) = x³. What is f⁻¹(27)?
4 of 8
f(x) = x² on x ≥ 0. What is f⁻¹(49)?
5 of 8
f(f⁻¹(9)) equals what?
6 of 8
The range of f is y ≥ 4. Smallest value in the domain of f⁻¹?
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
f(x) = 2x + 3. What is f⁻¹(9)?
Question 2 of 2
Why must a function be one-to-one to have an inverse?
What You Learned
- An inverse exists exactly when the function is one-to-one.
- Swap x and y, then solve for y.
- Verify by composing in both directions.