Page 2: Advanced Inverse Function Concepts

This page delves deeper into finding inverse functions for more complex equations and explores their graphical representations.

The page begins with an example of finding the inverse of f(x) = √(3x + 1), demonstrating the step-by-step algebraic process.

Another example shows how to find the inverse of g(x) = (x - 3) / (x + 2), which involves more complex algebraic manipulation.

Highlight: A function only has an inverse if it is a one-to-one function, meaning for every input there is exactly one output.

The page concludes with a discussion on graphing inverse functions:

Definition: To find the graph of y = f^(-1)(x), reflect the graph of y = f(x) in the diagonal line y = x.

A visual representation is provided to illustrate this concept, showing the original function and its inverse reflected across the line y = x.

Example: The graph shows how the original function and its inverse are symmetrical about the line y = x.

This comprehensive guide provides students with a solid foundation in understanding and working with inverse functions in higher mathematics.

Page 1: Introduction to Inverse Functions

This page introduces the concept of inverse functions and provides methods for identifying and finding them.

Definition: Inverse functions are functions that reverse the effect of each other.

The identity function is used to demonstrate that two functions are inverse functions:

Example: f(g(x)) = g(f(x)) = x

A step-by-step example is provided to show that f(x) = 4x-1 and g(x) = (x+1)/4 are inverse functions.

The page then outlines the algebraic method for finding inverse functions:

1. Write f(x) as y = ...
2. Rearrange to make x the subject
3. Swap x and y
4. Write as f^(-1)(x) = ...

Highlight: The inverse of f(x) is denoted as f^(-1)(x).

An example is given to find the inverse of f(x) = 5x + 4 and determine its domain.

Vocabulary: Domain refers to the set of possible input values for a function.