Inverse Functions
Learn how to find inverse functions algebraically, use the horizontal line test, and understand the graphical relationship between a function and its inverse.
What is an Inverse Function?
The inverse function f-1(x) "undoes" what f(x) does. If f takes input a to output b, then f-1 takes input b back to output a.
If f(a) = b, then f-1(b) = a
Also: f(f-1(x)) = x and f-1(f(x)) = x
Important: f-1(x) does not mean 1/f(x). The superscript -1 denotes the inverse function, not a reciprocal.
Not every function has an inverse. A function must be one-to-one (each y-value corresponds to exactly one x-value) to have an inverse function.
Finding Inverse Functions
To find f-1(x) algebraically:
- Write y = f(x).
- Swap x and y (interchange the variables).
- Solve for y in terms of x.
- Write the result as f-1(x) = ...
Horizontal Line Test
A function has an inverse if and only if every horizontal line crosses the graph at most once.
This confirms the function is one-to-one.
Graphing the Inverse
The graph of f-1 is the reflection of f in the line y = x.
Every point (a, b) on f becomes (b, a) on f-1.
Domain and Range of Inverse Functions
When finding the inverse, the domain and range swap:
Original function f
Domain of f → Range of f-1
Inverse function f-1
Range of f → Domain of f-1
For non-one-to-one functions (like y = x²), you can restrict the domain to make the function one-to-one, and then find the inverse on that restricted domain. For example, y = x² with x ≥ 0 has inverse y = √x.
Key Vocabulary
Inverse Function
A function f-1 that reverses the mapping of f, so that f-1(f(x)) = x.
One-to-One
A function where each output value corresponds to exactly one input. Required for an inverse to exist.
Horizontal Line Test
A graphical test: if every horizontal line intersects the graph at most once, the function is one-to-one.
Reflection
The graph of f-1 is the mirror image of f across the line y = x.
Worked Examples
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 for x ≥ 0.
Step 1: y = x² + 1. Swap: x = y² + 1.
Step 2: Solve: y² = x - 1, so y = ±√(x - 1).
Step 3: Since the original domain is x ≥ 0, the range of f is y ≥ 1, and we take the positive root.
Answer: f-1(x) = √(x - 1), for x ≥ 1.
Verify that f(x) = 2x + 3 and g(x) = (x - 3)/2 are inverses.
Check f(g(x)): f((x-3)/2) = 2 × (x-3)/2 + 3 = (x-3) + 3 = x. ✓
Check g(f(x)): g(2x+3) = ((2x+3)-3)/2 = 2x/2 = x. ✓
Answer: Since f(g(x)) = x and g(f(x)) = x, f and g are inverses of each other.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the inverse of f(x) = 5x + 2?
Question 2
The graph of f-1 is the reflection of f across which line?
Question 3
If f(4) = 7, what is f-1(7)?
Question 4
Which test determines if a function has an inverse?
Question 5
If the domain of f is [0, ∞) and the range of f is [3, ∞), what is the domain of f-1?
Key Concepts Summary
- ●An inverse function f-1 reverses the mapping of f: if f(a) = b then f-1(b) = a.
- ●To find the inverse: swap x and y, then solve for y.
- ●The horizontal line test determines if a function is one-to-one and has an inverse.
- ●The graph of f-1 is the reflection of the graph of f in the line y = x.
- ●The domain of f-1 equals the range of f, and the range of f-1 equals the domain of f.