Inverse Functions
Learn how to find the inverse of a function algebraically, understand how domain and range swap, and interpret inverse functions graphically.
What Is an Inverse Function?
The inverse of a function f(x) is denoted f−1(x). It reverses the action of f. If f maps x to y, then f−1 maps y back to x.
Key Property
f(f−1(x)) = x and f−1(f(x)) = x
Applying a function and then its inverse returns the original input
Example: f(x) = 2x + 3
f maps 4 → 11. The inverse maps 11 → 4.
f−1(x) = (x − 3) / 2
Everyday Example
Putting on shoes is one function. Taking them off is the inverse. The result of doing both is unchanged feet!
Finding the Inverse Algebraically
To find f−1(x) algebraically, follow these steps:
- Write y = f(x)
- Swap x and y (replace every x with y and every y with x)
- Solve the new equation for y
- Write f−1(x) = the expression for y
Find the inverse of f(x) = 3x − 5
Step 1: Write y = 3x − 5
Step 2: Swap x and y: x = 3y − 5
Step 3: Solve for y: x + 5 = 3y, so y = (x + 5) / 3
Answer: f−1(x) = (x + 5) / 3
Find the inverse of f(x) = x² + 1, x ≥ 0
Step 1: y = x² + 1
Step 2: Swap: x = y² + 1
Step 3: Solve: y² = x − 1, so y = √(x − 1) (taking positive root since x ≥ 0)
Answer: f−1(x) = √(x − 1), domain x ≥ 1
Verify: f(x) = 2x + 1 and f−1(x) = (x − 1) / 2
Check f(f−1(x)): f((x−1)/2) = 2 × (x−1)/2 + 1 = (x−1) + 1 = x ✓
Check f−1(f(x)): f−1(2x+1) = (2x+1−1)/2 = 2x/2 = x ✓
Domain, Range, and Graphs
Domain and Range Swap
The domain of f becomes the range of f−1, and vice versa. If (a, b) is on the graph of f, then (b, a) is on the graph of f−1.
The graph of f−1(x) is the reflection of the graph of f(x) in the line y = x.
Key Vocabulary
One-to-One Function
A function where each output corresponds to exactly one input. Only one-to-one functions have inverses that are also functions.
Horizontal Line Test
A function has an inverse if no horizontal line crosses its graph more than once. Used to check if a function is one-to-one.
Domain Restriction
Limiting the domain of a function so that it becomes one-to-one and its inverse is also a function.
f−1(x)
Notation for the inverse of f. Note: this is NOT the same as 1/f(x). The −1 is not an exponent here.
Knowledge Check
Test your understanding of inverse functions.
Question 1
What is the inverse of f(x) = x + 7?
Question 2
If f(x) = 4x − 8, what is f−1(x)?
Question 3
The graph of f−1(x) is the reflection of f(x) in which line?
Question 4
If f(3) = 10, what is f−1(10)?
Question 5
If the domain of f(x) is {x : x ≥ 2} and its range is {y : y ≥ 0}, what is the domain of f−1(x)?
Key Concepts Summary
- ●The inverse of f(x) reverses the mapping: if f(a) = b then f−1(b) = a.
- ●To find the inverse: swap x and y, then solve for y.
- ●The domain and range swap: domain of f−1 = range of f, and vice versa.
- ●The graph of f−1(x) is the reflection of f(x) in the line y = x.
- ●Only one-to-one functions have proper inverses (use the horizontal line test to check).