Definition of Inverse Functions
A function \$g\$ is the inverse of a function \$f\$ if and only if:
- \$f(g(x)) = x\$ for all \$x\$ in the domain of \$g\$.
- \$g(f(x)) = x\$ for all \$x\$ in the domain of \$f\$.
The inverse of \$f\$ is denoted by \$f^{-1}\$.
The notation \$f^{-1}\$ does not mean \$\frac{1}{f}\$. It specifically denotes the inverse function.
Finding the Inverse of a Function
To find the inverse of a function \$f(x)\$:
- Replace \$f(x)\$ with \$y\$.
- Swap \$x\$ and \$y\$.
- Solve for \$y\$.
- Replace \$y\$ with \$f^{-1}(x)\$.
Example 1:
Find the inverse of \$f(x) = 2x + 3\$.
- Replace \$f(x)\$ with \$y\$: \$y = 2x + 3\$.
- Swap \$x\$ and \$y\$: \$x = 2y + 3\$.
- Solve for \$y\$: \$y = \frac{x - 3}{2}\$.
- Replace \$y\$ with \$f^{-1}(x)\$: \$f^{-1}(x) = \frac{x - 3}{2}\$.
Verifying Inverse Functions
To verify that \$g(x)\$ is the inverse of \$f(x)\$, check that:
- \$f(g(x)) = x\$ for all \$x\$ in the domain of \$g\$.
- \$g(f(x)) = x\$ for all \$x\$ in the domain of \$f\$.
Example 2:
Verify that \$g(x) = \frac{x - 3}{2}\$ is the inverse of \$f(x) = 2x + 3\$.
- \$f(g(x)) = 2\left(\frac{x - 3}{2}\right) + 3 = x\$.
- \$g(f(x)) = \frac{2x + 3 - 3}{2} = x\$.
Both conditions are satisfied, so \$g(x)\$ is the inverse of \$f(x)\$.
Graphs of Inverse Functions
The graph of an inverse function is a reflection of the original function across the line \$y = x\$.
Example 3:
The graph of \$f(x) = 2x + 3\$ and its inverse \$f^{-1}(x) = \frac{x - 3}{2}\$ are reflections across the line \$y = x\$.
Not all functions have inverses. A function must be bijective (both injective and surjective) to have an inverse.
Inverses of Common Functions
Linear Functions
The inverse of a linear function \$f(x) = ax + b\$ is \$f^{-1}(x) = \frac{x - b}{a}\$.
Quadratic Functions
Quadratic functions are not bijective over their entire domain, so they do not have inverses unless their domain is restricted.
Exponential and Logarithmic Functions
Exponential functions and logarithmic functions are inverses of each other.
Example 4:
The inverse of \$f(x) = 2^x\$ is \$f^{-1}(x) = \log_2(x)\$.
- Find the inverse of \$f(x) = 3x - 7\$.
- Verify that \$g(x) = \frac{x + 7}{3}\$ is the inverse of \$f(x) = 3x - 7\$.
- Sketch the graph of \$f(x) = x^2\$ (restricted to \$x \geq 0\$) and its inverse.
How do we know that the inverse of a function is unique? What does this tell us about the nature of mathematical functions?
