FUNCTIONS 12
Inverse Functions
An inverse function reverses the action of the original function, exchanging inputs and outputs.
The main idea
If
\[f(a)=b,\]then the inverse function satisfies
\[\boxed{f^{-1}(b)=a}.\]The original function and its inverse describe the same relationship in opposite directions.
1. A simple inverse
If
\[f(x)=x+3,\]then the inverse must undo the addition:
\[\boxed{f^{-1}(x)=x-3}.\]↓ add 3
\(x+3\)
↓ subtract 3
\(x\)
2. Inverse notation
The notation \(f^{-1}(x)\) means the inverse function.
It does not mean \(1/f(x)\).
\(1/f\) = reciprocal
3. How to find an inverse
For \(f(x)=2x+5\), write \(y=2x+5\), solve for \(x\), then rename the variables:
\[\boxed{f^{-1}(x)=\frac{x-5}{2}}.\]↓ solve for \(x\)
\(x=(y-5)/2\)
↓
\(f^{-1}(x)=(x-5)/2\)
4. Checking an inverse
A correct inverse returns the original input:
\[f^{-1}(f(x))=x.\]then \(f^{-1}\)
↓
original input
5. Graphs of inverse functions
If \((a,b)\) lies on \(y=f(x)\), then \((b,a)\) lies on \(y=f^{-1}(x)\). The graphs are reflections in \(y=x\).
6. Not every function has an inverse
For \(f(x)=x^2\), both \(2\) and \(-2\) give \(4\). So \(x^2\) is not one-to-one on all real numbers.
\(2\to4\)
two inputs → one output
7. The horizontal line test
A function has an inverse only if every horizontal line meets its graph at most once.
8. Restricting the domain
Restricting \(f(x)=x^2\) to \(x\ge0\) makes it one-to-one, with
\[\boxed{f^{-1}(x)=\sqrt{x}}.\]9. Domain and range swap
\[\operatorname{Dom}(f)=\operatorname{Ran}(f^{-1}),\qquad\operatorname{Ran}(f)=\operatorname{Dom}(f^{-1}).\]range of \(f\) ↔ domain of \(f^{-1}\)
10. Exponential and logarithmic inverses
\(y=e^x\) and \(y=\ln x\) are inverses and reflect across \(y=x\).
11. Why inverse functions matter for calculus
Inverse functions lead to logarithms, inverse trigonometric functions and the derivative rule
\[\boxed{(f^{-1})'(x)=\frac{1}{f'(f^{-1}(x))}}.\]↓
related slopes
An inverse reverses a function. Its graph is reflected in \(y=x\), domain and range exchange roles, and the original function must be one-to-one on its chosen domain.