Finding Inverse Functions: Step-by-Step Guide
The inverse of a function f, written f^-1, undoes what f does. If f(a) = b, then f^-1(b) = a. To find it, write y for f(x), swap x and y, and solve for y. Only functions that pass the horizontal line test (no output is used twice) have an inverse.
Math Racer skill guide · Algebra 2 · Grade 11
How to solve it, step by step
- Replace f(x) with y. Write the equation using y instead of f(x).\[y = f(x)\]
- Swap x and y. Switch every x with y and every y with x. On a graph, this flips the picture over the line y = x.\[x = f(y)\]
- Solve for y. Use algebra to get y by itself.\[y = f^{-1}(x)\]
- Write it as f^-1(x). Replace y with f^-1(x) to name the inverse.\[f^{-1}(x) = \ldots\]
- Check with composition. Check that f(f^-1(x)) = x and f^-1(f(x)) = x. If both give x, you have the right inverse.\[f\bigl(f^{-1}(x)\bigr) = x \;\;\text{and}\;\; f^{-1}\bigl(f(x)\bigr) = x\]
Worked example
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. Add 5: x + 5 = 3y. Divide by 3: y = (x + 5) / 3.
Step 4: Write it with inverse notation: f^-1(x) = (x + 5) / 3.
Step 5: Check: f(f^-1(x)) = 3 x (x + 5)/3 - 5 = (x + 5) - 5 = x. And f^-1(f(x)) = (3x - 5 + 5)/3 = 3x/3 = x. Both work.
The inverse is f^-1(x) = (x + 5) / 3.
Tips and common mistakes
- Only one-to-one functions have an inverse. If a function fails the horizontal line test, limit its domain first.
- f^-1(x) does not mean 1/f(x). The -1 is not an exponent here. It names the inverse function.
- Swapping x and y is the key idea. Solving for y is the algebra that follows.
- The domain of the inverse is the range of f, and the range of the inverse is the domain of f.
- Always check with composition. Sign mistakes are easy to make.
- For a line f(x) = ax + b, the inverse is f^-1(x) = (x - b) / a.
- For x^2, limit the domain to x >= 0 so there is an inverse.