Functions and their inverses

Functions and Their Inverses

A function is a relation between a set of inputs and a set of possible outputs. Each input is related to exactly one output. In this section, you will learn about functions and how to find their inverses.

Understanding Functions

A function can be represented in various forms, such as a table, a graph, or an equation. The most common representation is through an equation, which describes the relationship between the input (often called x) and the output (often called y).

For example, consider the function defined by the equation:

y = 2x + 3

In this function, for every value of x, there is a corresponding value of y. Let’s calculate some values:

• If x = 1, then y = 2(1) + 3 = 5.
• If x = 2, then y = 2(2) + 3 = 7.
• If x = 0, then y = 2(0) + 3 = 3.

This function can be represented as a set of ordered pairs: {(1, 5), (2, 7), (0, 3)}.

Characteristics of Functions

Functions have specific characteristics that help us understand their behaviour:

• Domain: The set of all possible input values (x-values) for a function.
• Range: The set of all possible output values (y-values) for a function.
• Intercepts: Points where the function crosses the axes. The x-intercept occurs when y = 0, and the y-intercept occurs when x = 0.

Types of Functions

Functions can be classified into various types, such as:

• Linear Functions: Functions of the form y = mx + b, where m is the slope and b is the y-intercept.
• Quadratic Functions: Functions of the form y = ax^2 + bx + c, where a, b, and c are constants.
• Cubic Functions: Functions of the form y = ax^3 + bx^2 + cx + d.

Inverse Functions

An inverse function reverses the effect of the original function. If you have a function f(x), its inverse is denoted as f^-1(x). For example, if f(x) = y, then f^-1(y) = x.

To find the inverse of a function, follow these steps:

1. Replace f(x) with y.
2. Swap x and y in the equation.
3. Solve for y.
4. Replace y with f^-1(x).

Example: Finding the Inverse of a Function

Let’s find the inverse of the function f(x) = 2x + 3.

1. Replace f(x) with y: y = 2x + 3.
2. Swap x and y: x = 2y + 3.
3. Solve for y:

x - 3 = 2y

y = (x - 3)/2

4. Replace y with f^-1(x): f^-1(x) = (x - 3)/2.

The inverse function is f^-1(x) = (x - 3)/2.

Graphing Inverse Functions

The graph of an inverse function is a reflection of the original function across the line y = x. This means that if you take any point (a, b) on the graph of f(x), the point (b, a) will be on the graph of f^-1(x).

Example: Graphing a Function and Its Inverse

Let’s graph the function f(x) = 2x + 3 and its inverse f^-1(x) = (x - 3)/2.

Common Mistakes

Summary

• A function relates inputs to outputs, with each input having exactly one output.
• The inverse of a function reverses the original function’s effect.
• To find the inverse, swap x and y, then solve for y.
• The graph of an inverse function is a reflection of the original function across the line y = x.

Check your understanding

1. Determine the inverse of the function f(x) = 3x - 4.
2. What is the domain and range of the function g(x) = x^2?
3. Graph the function h(x) = -x + 5 and its inverse. What do you notice about the relationship between the two graphs?
4. Explain why not all functions have inverses.