Lesson 3: Inverse Functions Revisited

Property

If $f(x)$ is a one-to-one function whose ordered pairs are of the form $(x, y)$, then its inverse function $f^{-1}(x)$ is the set of ordered pairs $(y, x)$. The graphs of $f$ and $f^{-1}$ are mirror images of each other through the line $y=x$.

Examples

Property

To find the inverse of a one-to-one function algebraically:
Step 1. Substitute $y$ for $f(x)$.
Step 2. Interchange the variables $x$ and $y$.
Step 3. Solve for $y$.
Step 4. Substitute $f^{-1}(x)$ for $y$.

Examples

• Find the inverse of $f(x) = 3x - 5$. Step 1: $y = 3x - 5$. Step 2: $x = 3y - 5$. Step 3: $x+5 = 3y$, so $y = \frac{x+5}{3}$. Step 4: $f^{-1}(x) = \frac{x+5}{3}$.

• Find the inverse of $f(x) = \sqrt[3]{x-1}$. Step 1: $y = \sqrt[3]{x-1}$. Step 2: $x = \sqrt[3]{y-1}$. Step 3: $x^3 = y-1$, so $y = x^3+1$. Step 4: $f^{-1}(x) = x^3+1$.

Property

To graph an inverse function $f^{-1}(x)$, reflect the graph of $f(x)$ across the line $y = x$ by swapping the $x$ and $y$ coordinates of each point. If $(a, b)$ is on the graph of $f(x)$, then $(b, a)$ is on the graph of $f^{-1}(x)$.

Examples