IB Mathematics AA SL Function and their domain range graph Study Notes

Functions, Domain, Range, and Graphs

Functions

A function is a mathematical relationship that assigns each input value (domain) to exactly one output value (range).

Example: Let \( f(x) = x^2 \). For \( x = 2 \), the output is \( f(2) = 4 \). For \( x = -3 \), the output is \( f(-3) = 9 \).

Domain

The domain of a function is the set of all possible input values. For example, the domain of \( f(x) = \sqrt{x} \) is \( x \ge 0 \) because square roots of negative numbers are not defined in real numbers.

Range

The range of a function is the set of all possible output values. For instance, the range of \( f(x) = x^2 \) is \( f(x) \ge 0 \) since squares of real numbers are non-negative.

Types of Functions

• One-to-One Function: Each input value maps to a unique output value. For example, the function \( f(x) = 2x + 3 \) is one-to-one. The graph passes the horizontal line test.

• Many-to-One Function: Multiple input values can map to the same output value. For example, \( f(x) = x^2 \) is a many-to-one function because both \( x = 2 \) and \( x = -2 \) give the same output, \( f(x) = 4 \).

Inverse Function

An inverse function reverses the operation of a given function. If \( f \) is a function with an inverse, denoted as \( f^{-1} \), then \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \).

Example: Let \( f(x) = 2x + 3 \). To find its inverse, follow these steps:

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

\( x – 3 = 2y \)

\( y = \frac{x – 3}{2} \)

1. Thus, the inverse function is \( f^{-1}(x) = \frac{x – 3}{2} \).

Key Note: For a function to have an inverse, it must be a one-to-one function (pass the horizontal line test).

Graphing Functions and Inverses

The graph of an inverse function is a reflection of the graph of the original function across the line \( y = x \).

Example Problems on Inverse Functions

Example 1: Verifying an Inverse Function

Problem: Verify if \( g(x) = \frac{x – 3}{2} \) is the inverse of \( f(x) = 2x + 3 \).

Solution:

We need to check if \( f(g(x)) = x \) and \( g(f(x)) = x \).

\( f(g(x)) = 2\left(\frac{x – 3}{2}\right) + 3 = x – 3 + 3 = x \)

\( g(f(x)) = \frac{2x + 3 – 3}{2} = \frac{2x}{2} = x \)

Answer: Yes, \( g(x) \) is the inverse of \( f(x) \).

Example 2: Inverse of a Quadratic Function (Restricted Domain)

Problem: Find the inverse of \( f(x) = x^2 \) with the restricted domain \( x \ge 0 \).

Solution:

Replace \( f(x) \) with \( y \): \( y = x^2 \).

Swap \( x \) and \( y \): \( x = y^2 \).

Solve for \( y \): \( y = \sqrt{x} \).

Since the domain is restricted to \( x \ge 0 \), the inverse is:

\( f^{-1}(x) = \sqrt{x} \) for \( x \ge 0 \).

Answer: The inverse function is \( f^{-1}(x) = \sqrt{x} \).

Key Points

• Domain and Range of Inverse: The domain of the inverse function is the range of the original function, and vice versa.
• Horizontal Line Test: A function must be one-to-one to have an inverse. The graph of a function must pass the horizontal line test.

Remember:

• One-to-One Function Requirement: Only one-to-one functions have inverses that are also functions.
• Graph Reflection: The graph of a function and its inverse are symmetric across the line \( y = x \).