Inverse Function Formula
The Formula
When to use: If f turns a into b, then f^{-1} turns b back into a. Reverse the process.
Quick Example
f^{-1}(x) = \frac{x}{2} (halve).
f^{-1}(f(3)) = f^{-1}(6) = 3.
Notation
What This Formula Means
The inverse of a function f is a function f^{-1} that reverses f: if f(a) = b then f^{-1}(b) = a. It exists only when f is one-to-one.
If f turns a into b, then f^{-1} turns b back into a. Reverse the process.
Formal View
Worked Examples
Example 1
easySolution
- 1 Write y = 3x - 7.
- 2 Swap x and y: x = 3y - 7.
- 3 Solve for y: 3y = x + 7, so y = \frac{x + 7}{3}.
- 4 Therefore f^{-1}(x) = \frac{x + 7}{3}.
Answer
Example 2
mediumCommon Mistakes
- Thinking f^{-1}(x) = \frac{1}{f(x)} — the -1 superscript means inverse function, NOT reciprocal
- Trying to find the inverse of a non-one-to-one function — f(x) = x^2 has no inverse on all reals because f(2) = f(-2) = 4
- Forgetting to swap x and y when finding the inverse algebraically — solve x = f(y) for y, don't just manipulate f(x)
Why This Formula Matters
Inverse functions are how we solve equations — finding x such that f(x) = b is exactly computing f^{-1}(b). Logarithm is the inverse of exponential.
Frequently Asked Questions
What is the Inverse Function formula?
The inverse of a function f is a function f^{-1} that reverses f: if f(a) = b then f^{-1}(b) = a. It exists only when f is one-to-one.
How do you use the Inverse Function formula?
If f turns a into b, then f^{-1} turns b back into a. Reverse the process.
What do the symbols mean in the Inverse Function formula?
f^{-1} denotes the inverse function. To find it: write y = f(x), swap x and y, solve for y.
Why is the Inverse Function formula important in Math?
Inverse functions are how we solve equations — finding x such that f(x) = b is exactly computing f^{-1}(b). Logarithm is the inverse of exponential.
What do students get wrong about Inverse Function?
f^{-1}(x) is NOT \frac{1}{f(x)}. The -1 means inverse, not reciprocal.
What should I learn before the Inverse Function formula?
Before studying the Inverse Function formula, you should understand: function definition.
Want the Full Guide?
This formula is covered in depth in our complete guide:
Functions and Graphs: Complete Foundations for Algebra and Calculus →