An inverse undoes the function: it takes the output back to the input it came from.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Finding one
Swap x and y, then solve for y. That is the whole procedure.
On the graph
The inverse is the reflection across the line y = x, because swapping the coordinates is exactly that reflection.
When it exists
Only a one-to-one function has an inverse. x² does not, because 4 could have come from 2 or −2.
Only one-to-one functions can be undone
If two inputs share an output, the inverse would need to send one input to two places. The horizontal line test decides it, and restricting the domain rescues functions that fail.
Swap and solve
Exchange x and y in the equation, then solve for y. The swap is the statement that inputs and outputs have exchanged roles, so it is the definition rather than a trick.
Reflection in y = x
The graph of the inverse is the original reflected in the line y = x, and domain and range exchange. A point (a, b) on f corresponds to (b, a) on f⁻¹, which is a quick way to sketch the inverse.
Monotonic functions are always invertible
A function that is strictly increasing or strictly decreasing throughout its domain passes the horizontal line test automatically. That is why exponentials and logarithms are inverses without any restriction.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Priya Invert a Linear Function
Priya inverts f(x) = 3x + 6.
- Step 1
She writes y = 3x + 6, then swaps: x = 3y + 6.
Step 4: Your Turn
Practice makes it stick.
Undoing
Problem 1 of 2
f(x) = x + 8. What does the inverse give when the input is 20?
The Round Trip
Problem 2 of 2
f(x) = 5x. What is f applied to the inverse of 45?
Swap and Solve
1 of 8
f(x) = x − 4. Inverse of 10?
2 of 8
f(x) = 2x. Inverse of 18?
3 of 8
f(x) = 3x − 1. Inverse of 14?
4 of 8
Select every function that has an inverse over all real numbers.
5 of 8
Across which line is an inverse a reflection? Answer 1 for y = x, 2 for the x-axis.
6 of 8
The point (3, 11) is on f. What is the x of the matching point on the inverse?
7 of 8
Order the steps for finding an inverse.
- 1Swap x and y
- 2Solve for y
- 3Check with one point
- 4Write y = f(x)
8 of 8
Why has x² no inverse over all reals? Answer 1 if two inputs share an output, 0 otherwise.
Step 5: Quick Check
Show what you know.
Question 1 of 1
f(x) = 4x + 2. Inverse of 22?
What You Learned
- An inverse sends each output back to the input it came from.
- Find it by swapping x and y, then solving for y.
- Only a one-to-one function has an inverse, and the graph is a reflection in y = x.