Functions and Graphs

Inverse Functions

Inverse functions are a fundamental concept in algebra and precalculus, allowing us to 'reverse' the effect of a function. Understanding how to verify, find, and graph inverse functions is essential for solving equations and modeling real-world phenomena.

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 the range of f-1, and the range of f is the domain of f-1.

Verifying Inverse Functions

• To verify that two functions f and g are inverses, check that:

  • for all x in the domain of g

  • for all x in the domain of f

• Example: If f(x) = 2x + 3 and g(x) = (x - 3)/2, then:

Finding the Inverse of a Function

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, then f has an inverse function.

4. If f has an inverse, replace y with f^{-1}(x).

• Example: Find the inverse of f(x) = 3x - 5:

  • Step 1:

  • Step 2: Interchange and :

  • Step 3: Solve for :

  • Step 4:

The Horizontal Line Test for Inverse Functions

• A function f has an inverse that is also a function if and only if no horizontal line intersects the graph of f more than once.

• This test determines if f is one-to-one.

• Example: The function f(x) = x^2 fails the horizontal line test (unless restricted to ), so it does not have an inverse function over all real numbers.

Graphs of f and f-1

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

• To graph both functions, plot f, then reflect each point to for f^{-1}.

Example: Graphing the Inverse Function

• Suppose f consists of two line segments: one from to , and another from $(2, 3)$ to .

• The inverse function will have segments from to and from $(3, 2)$ to .

• Plot these points and connect them to visualize the inverse.

Summary Table: Steps for Finding and Verifying Inverse Functions