Inverse Functions: Definitions, Properties, and Graphical Analysis

Functions and Graphs

Inverse Functions

Inverse functions are a fundamental concept in College Algebra, allowing us to reverse the effect of a function. This section covers how to verify, find, and graph inverse functions, as well as how to determine if a function has an inverse using the horizontal line test.

Definition of the Inverse of a Function

• Inverse Function: Let f and g be two functions. If for every x in the domain of g, f(g(x)) = x, and for every x in the domain of f, g(f(x)) = x, then g is the inverse of f, denoted as f^{-1}.

• The domain of f is equal to the range of f^{-1}, and vice versa.

Verifying Inverse Functions

To verify that two functions are inverses, 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: If f(x) = 2x + 3 and g(x) = \frac{x - 3}{2}, then:

• $f(g(x)) = 2\left(\frac{x - 3}{2}\right) + 3 = x - 3 + 3 = x$

• $g(f(x)) = \frac{2x + 3 - 3}{2} = \frac{2x}{2} = x$

Finding the Inverse of a Function

To find the inverse of a function f:

1. Replace f(x) with y in the equation for f.

2. Interchange x and y.

3. Solve for y. If this equation defines y as a function of x, f has an inverse.

4. Replace y with f^{-1}(x).

Example: Find the inverse of f(x) = 3x - 4:

• Step 1: $y = 3x - 4$

• Step 2: $x = 3y - 4$

• Step 3: $x + 4 = 3y \Rightarrow y = \frac{x + 4}{3}$

• Step 4: $f^{-1}(x) = \frac{x + 4}{3}$

The Horizontal Line Test for Inverse Functions

The horizontal line test determines if a function has an inverse that is also a function:

• If no horizontal line intersects the graph of f at more than one point, then f has an inverse function.

Example: A graph that passes the horizontal line test represents a one-to-one function and thus has an inverse.

Graphs of f and f Inverse

The graph of f^{-1} is a reflection of the graph of f about the line y = x.

• Each point (a, b) on the graph of f corresponds to (b, a) on the graph of f^{-1}.

Example: If f contains the point (1, 2), then f^{-1} contains the point (2, 1).

Graphing the Inverse Function

To graph the inverse function:

1. Plot the points of f.

2. Reverse the ordered pairs to plot f^{-1}.

3. Connect the points and verify symmetry about y = x.