Overview
This lecture explains the concept of inverse functions, how to find them for sets of coordinates and equations, and provides the step-by-step process for solving inverse functions.
Definition of Inverse Functions
- The inverse function is formed by exchanging the independent variable (x) with the dependent variable (y) in a relation.
- Independent variable (x) corresponds to the horizontal axis; dependent variable (y) corresponds to the vertical axis.
- Interchanging x and y gives the inverse relation.
- The inverse of a function is not always a function; check if each x-coordinate in the inverse is unique.
Example: Inverse of a Set of Coordinates
- To find the inverse, swap each (x, y) to (y, x).
- Example: (2, 0), (3, -1), (4, 0), (5, 4) becomes (0, 2), (-1, 3), (0, 4), (4, 5).
- If an x-value repeats in the inverse, it is not a function.
Steps to Find the Inverse of an Equation
- Replace f(x) with y.
- Interchange x and y in the equation.
- Solve the new equation for y.
- Replace y with f⁻¹(x) to indicate the inverse.
Example 1: Inverse of f(x) = x - 4
- Step 1: y = x - 4.
- Step 2: x = y - 4.
- Step 3: Solve for y: y = x + 4.
- Step 4: f⁻¹(x) = x + 4.
Example 2: Inverse of f(x) = 2x + 3
- Step 1: y = 2x + 3.
- Step 2: x = 2y + 3.
- Step 3: Solve for y: y = (x - 3)/2.
- Step 4: f⁻¹(x) = (x - 3)/2.
Key Terms & Definitions
- Inverse Function — a function formed by swapping the x and y variables of the original function.
- Dependent Variable — typically y, whose value depends on x.
- Independent Variable — typically x, the input of a function.
Action Items / Next Steps
- Practice finding inverses of functions using the four-step process.
- Try finding the inverse of additional equations for more practice.