Inverse, Exponential, and Logarithmic Functions

4.1 Inverse Functions

Inverse functions are essential in algebra for reversing the effect of a function. This section covers one-to-one functions, the definition and properties of inverse functions, and methods for finding and verifying inverses.

• One-to-One Functions: A function is one-to-one if each output value corresponds to only one input value. Formally, if implies , then is one-to-one.

• Horizontal Line Test: A function is one-to-one if every horizontal line intersects its graph at most once.

• Inverse Function Definition: If is a one-to-one function, its inverse satisfies:

  • for every in the domain of

  • for every in the domain of

• Finding Inverses: To find the inverse of :

  1. Replace with

  2. Interchange and

  3. Solve for

  4. Rewrite as

• Graphing Inverses: The graph of is a reflection of the graph of across the line .

Example: If , then .

4.2 Exponential Functions

Exponential functions model rapid growth or decay and are fundamental in many real-world applications. This section introduces their properties, graphs, and equations.

• Definition: An exponential function has the form , where and .

• Domain and Range:

  • Domain:

  • Range: for

• Graphing: The graph passes through and has a horizontal asymptote at .

• Reflections and Translations: Transformations such as reflect the graph across the -axis.

• Solving Exponential Equations: If , then .

Example: is increasing, is decreasing.

Compound Interest

Compound interest is a key application of exponential functions in finance. It models the growth of investments over time.

• Formula: , where:

  • = final amount

  • = principal

  • = annual interest rate

  • = number of compounding periods per year

  • = number of years

• Continuous Compounding:

Example: If , , , , then .

4.3 Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve equations involving exponents.

• Definition: For , , is the exponent to which must be raised to get .

• Properties:

  • Product:

  • Quotient:

  • Power:

• Change-of-Base Formula:

• Common Logarithm:

• Natural Logarithm:

Example: because .

Solving Logarithmic Equations

Logarithmic equations can be solved by rewriting them in exponential form or using properties of logarithms.

• If , then .

• Use properties to combine or expand logarithmic expressions before solving.

Example: implies , so .

4.4 Evaluating Logarithms and the Change-of-Base Theorem

This section covers how to evaluate logarithms using calculators and the change-of-base theorem.

• Change-of-Base Theorem:

• Use this formula to compute logarithms with any base using common or natural logarithms.

Example:

4.5 Exponential and Logarithmic Equations

Equations involving exponentials and logarithms can be solved using their properties and by converting between forms.

• To solve , set .

• To solve , set .

• Use logarithms to solve for exponents: .

Example:

4.6 Applications and Models of Exponential Growth and Decay

Exponential growth and decay models describe processes that increase or decrease at rates proportional to their current value.

• Exponential Growth: , where

• Exponential Decay: , where

• Doubling Time:

• Half-Life:

Example: If a population doubles every 5 years, .

Tables

Below is a summary table of logarithm properties:

Additional info: These notes include worked examples, step-by-step solutions, and graphical illustrations to reinforce understanding of inverse, exponential, and logarithmic functions, as well as their applications in finance and science.