Inverse Functions

Definition and Existence

An inverse function \(f^{-1}(x)\) “undoes” the action of the function \(f(x)\).
Formally, if \(y = f(x)\), then \(x = f^{-1}(y)\).
The inverse function exists if and only if the original function is one-to-one (i.e., it passes the horizontal line test).
A one-to-one function is also called an injective function.
If a function is not one-to-one, its domain can be restricted to make it one-to-one, allowing an inverse to be defined on that restricted domain.

Replace \(f(x)\) with \(y\).
Swap \(x\) and \(y\).
Solve for \(y\) in terms of \(x\). The resulting expression is \(f^{-1}(x)\).
State the domain and range of \(f^{-1}(x)\). Note that:
- Domain of \(f^{-1}(x)\) = Range of \(f(x)\)
- Range of \(f^{-1}(x)\) = Domain of \(f(x)\)

The graph of \(f^{-1}(x)\) is the reflection of the graph of \(f(x)\) across the line \(y = x\).

If \(f^{-1}(x)\) is the inverse of \(f(x)\), then:
- \(f(f^{-1}(x)) = x\) for all \(x\) in the domain of \(f^{-1}(x)\)
- \(f^{-1}(f(x)) = x\) for all \(x\) in the domain of \(f(x)\)

Function	Inverse Function	Domain of Original Function	Range of Original Function	Domain of Inverse Function	Range of Inverse Function
\(y = e^x\)	\(y = \ln(x)\)	\(\mathbb{R}\)	\((0, \infty)\)	\((0, \infty)\)	\(\mathbb{R}\)
\(y = a^x, a>0, a \ne 1\)	\(y = \log_a(x)\)	\(\mathbb{R}\)	\((0, \infty)\)	\((0, \infty)\)	\(\mathbb{R}\)
\(y = \sin(x)\)	\(y = \arcsin(x)\)	\([-\frac{\pi}{2}, \frac{\pi}{2}]\)	\([-1, 1]\)	\([-1, 1]\)	\([-\frac{\pi}{2}, \frac{\pi}{2}]\)
\(y = \cos(x)\)	\(y = \arccos(x)\)	\([0, \pi]\)	\([-1, 1]\)	\([-1, 1]\)	\([0, \pi]\)
\(y = \tan(x)\)	\(y = \arctan(x)\)	\((-\frac{\pi}{2}, \frac{\pi}{2})\)	\(\mathbb{R}\)	\(\mathbb{R}\)	\((-\frac{\pi}{2}, \frac{\pi}{2})\)
\(y = x^n\) (n odd)	\(y = x^{\frac{1}{n}}\)	\(\mathbb{R}\)	\(\mathbb{R}\)	\(\mathbb{R}\)	\(\mathbb{R}\)
\(y = x^n\) (n even, \(x \ge 0\))	\(y = x^{\frac{1}{n}}\)	\([0, \infty)\)	\([0, \infty)\)	\([0, \infty)\)	\([0, \infty)\)

Inverse functions can be used to solve equations involving exponential, logarithmic, circular (trigonometric), and power functions.
Example: Solve \(e^x = 5\).
- Take the natural logarithm of both sides: \(\ln(e^x) = \ln(5)\).
- Simplify: \(x = \ln(5)\).
Example: Solve \(\sin(x) = 0.5\) for \(x \in [-\frac{\pi}{2}, \frac{\pi}{2}]\).
- Take the inverse sine of both sides: \(\arcsin(\sin(x)) = \arcsin(0.5)\).
- Simplify: \(x = \arcsin(0.5) = \frac{\pi}{6}\).

Domain Restrictions: Be mindful of domain restrictions when dealing with inverse trigonometric functions and power functions with even exponents.
Multiple Solutions: Trigonometric equations often have multiple solutions. Use the domain restriction of the inverse function to find the principal solution, and then use trigonometric identities and the symmetry of the graphs to find other solutions within a specified interval.
Logarithmic Functions: Remember that the argument of a logarithmic function must be positive.
Exponential Functions: Exponential functions are always positive.