Finding the Inverse of a Function
Finding the Inverse of a Function is an algebra skill from Openstax Intermediate Algebra 2E, Chapter 10: Exponential and Logarithmic Functions. To find the inverse algebraically, students replace f(x) with y, interchange x and y, solve for y, and then replace y with f-inverse(x). This four-step process yields the inverse function, which reverses the original function by swapping inputs and outputs.
Key Concepts
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: $f^{ 1}(x) = x^3+1$.
Common Questions
What are the steps to find the inverse of a function algebraically?
Replace f(x) with y, interchange x and y, solve the resulting equation for y, then replace y with f-inverse(x).
Why do you swap x and y to find the inverse?
The inverse function reverses the role of input and output; swapping x and y mathematically reflects this reversal, turning outputs into inputs and vice versa.
How do you verify that two functions are inverses of each other?
Compute f(f-inverse(x)) and f-inverse(f(x)); if both compositions return x for all values in the respective domains, the functions are inverses.
What textbook covers finding the inverse of a function?
Openstax Intermediate Algebra 2E, Chapter 10: Exponential and Logarithmic Functions covers the algebraic method for finding inverse functions.