Inverse Functions- Everything You Need to Know
What Inverse Functions Actually Are
An inverse function is what you get when you reverse a function. If f takes x to y, the inverse function f-1 takes y back to x. That's the whole idea.
Think of it like a two-way street. f(x) is the forward direction. f-1(x) is the way back.
Not every function has an inverse. Only one-to-one functions pass the horizontal line test. If a horizontal line crosses your graph more than once, there's no inverse function.
The Formal Definition
If f is a one-to-one function, then f-1 exists and satisfies:
f(f-1(x)) = x and f-1(f(x)) = x
These two equations are the tests. If either one fails, they're not inverses.
How to Find an Inverse Function
Step-by-Step Process
Finding an inverse is just algebra. Here's how:
- Replace f(x) with y
- Swap x and y
- Solve for y
- Replace y with f-1(x)
That's it. Four steps.
Example: Finding f-1(x)
Given f(x) = 3x + 7
Step 1: y = 3x + 7
Step 2: x = 3y + 7
Step 3: x - 7 = 3y, so y = (x - 7)/3
Step 4: f-1(x) = (x - 7)/3
Example 2: A Quadratic
Given f(x) = x2 + 4
This function has no inverse unless you restrict the domain. Why? Because f(2) = 8 and f(-2) = 8. Two inputs give the same output.
Restrict the domain to x ≥ 0, and now f-1(x) = √(x - 4). Restrict to x ≤ 0, and f-1(x) = -√(x - 4).
Domain restriction is non-negotiable when working with quadratics, absolute value, and other non-one-to-one functions.
Domain and Range Swap
The domain of f becomes the range of f-1. The range of f becomes the domain of f-1.
Example: f(x) = 2x + 3 has domain (-∞, ∞) and range (-∞, ∞). f-1(x) = (x - 3)/2 also has domain (-∞, ∞) and range (-∞, ∞).
But for f(x) = x2 + 4 (with domain x ≥ 0), the domain is [0, ∞) and range is [4, ∞). The inverse f-1(x) = √(x - 4) has domain [4, ∞) and range [0, ∞).
Graphical Relationship
The graph of f-1 is a reflection of f across the line y = x.
This is useful. If you have a graph of f and need to sketch f-1, just flip it over the 45-degree line.
The point (a, b) on f becomes (b, a) on f-1.
Common Inverse Function Pairs
| Function f(x) | Inverse f-1(x) |
|---|---|
| mx + b | (x - b)/m |
| x2 (x ≥ 0) | √x |
| x2 (x ≤ 0) | -√x |
| 1/x | 1/x |
| √x | x2 (x ≥ 0) |
| ex | ln(x) |
| ln(x) | ex |
| sin(x) (restricted) | arcsin(x) |
Verifying Inverse Functions
Always verify. Plug in a value and check both composite functions.
Example: Verify that f(x) = 5x - 2 and g(x) = (x + 2)/5 are inverses.
f(g(x)) = 5((x + 2)/5) - 2 = x + 2 - 2 = x ✓
g(f(x)) = (5x - 2 + 2)/5 = 5x/5 = x ✓
Both compositions equal x. They're inverses. If either one fails, they're not.
Trigonometric Function Inverses
Trig functions need restricted domains to have inverses. Here's what you need to know:
- arcsin(x) has domain [-1, 1] and range [-π/2, π/2]
- arccos(x) has domain [-1, 1] and range [0, π]
- arctan(x) has domain (-∞, ∞) and range (-π/2, π/2)
These ranges are standard. Memorize them.
Getting Started: Your Inverse Function Checklist
Before you declare that an inverse exists:
- Check if the function is one-to-one. Horizontal line test. If it fails, restrict the domain.
- Find the inverse algebraically. Replace f(x) with y, swap x and y, solve for y.
- State the domain and range of the inverse. They're swapped from the original.
- Verify. Compute f(f-1(x)) and f-1(f(x)). Both must equal x.
- Graph if needed. Reflect across y = x.
Where Students Go Wrong
Forgetting domain restrictions. This is the #1 mistake. Every non-one-to-one function needs a restricted domain before you can find its inverse.
Confusing f-1(x) with 1/f(x). These are completely different. f-1(x) means inverse function. (f(x))-1 means reciprocal.
Not verifying. Always check your work. One composite function failing means you made an error.
Overcomplicating it. Inverse functions are about undoing operations. If f(x) adds 5, f-1(x) subtracts 5. Keep it simple.