Exponential and Logarithmic Functions

One-to-One Functions and Inverse Functions

This section explores the concept of one-to-one functions, how to determine if a function is one-to-one, and the process of finding and verifying inverse functions. These concepts are foundational for understanding exponential and logarithmic functions in precalculus.

One-to-One Functions

Definition and Identification

A one-to-one function is a function in which every element of the range is paired with exactly one element of the domain. In other words, no two different inputs produce the same output.

• Formal Definition: A function f is one-to-one if, for any two different inputs x1 and x2, f(x1) \neq f(x2).

• Example: If two different ages correspond to the same cholesterol score, the function is not one-to-one.

The Horizontal Line Test

The Horizontal Line Test is a graphical method to determine if a function is one-to-one. If every horizontal line intersects the graph of a function at most once, the function is one-to-one.

• If a horizontal line crosses the graph more than once, the function is not one-to-one.

Examples Using the Horizontal Line Test

• Example 1: The function y = x^6 is not one-to-one because the horizontal line y = 1 intersects the graph at two points: (-1, 1) and (1, 1).

• Example 2: The function y = \sqrt[3]{x} is one-to-one because every horizontal line intersects the graph exactly once.

Monotonic Functions

• A function that is increasing or decreasing on an interval is one-to-one on that interval.

Inverse Functions

Definition and Notation

If f is a one-to-one function, its inverse function is denoted by f^{-1}. The inverse function reverses the correspondence of f: it maps each output of f back to its original input.

• Notation: If f(a) = b, then f^{-1}(b) = a.

Finding the Inverse from a Map or Set of Ordered Pairs

• To find the inverse, interchange the domain and range elements.

After interchanging, the inverse function is:

Domain and Range of Inverse Functions

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

Graphing Inverse Functions

Symmetry with Respect to y = x

The graphs of a one-to-one function and its inverse are symmetric with respect to the line y = x.

Example: Graphing a Function and Its Inverse

• Given a function f with points (-2, 4), (-1/2, 2), and (3, 3), the inverse function will have points (4, -2), (2, -1/2), and (3, 3).

Verifying Inverse Functions

Definition

Two functions f and g are inverses if:

• for all x in the domain of g

• for all x in the domain of f

Finding the Inverse of a Function Defined by an Equation

Procedure

1. Replace f(x) with y.

2. Interchange x and y in the equation.

3. Solve for y to obtain the explicit form of the inverse function.

4. Check by verifying and .

Example: Linear Function

• Given , find the inverse.

• Step 1: Replace with :

• Step 2: Interchange and :

• Step 3: Solve for :

Example: Domain-Restricted Function

• Given with , find the inverse.

• Step 1: Replace with :

• Step 2: Interchange and :

• Step 3: Solve for : (since )

Summary Table: Properties of Inverse Functions