Chapter 1: Functions

Overview

Functions form the cornerstone of Additional Mathematics and are essential for understanding more advanced mathematical concepts. This chapter explores the fundamental principles of functions, including their definitions, classifications, properties, and applications. Students will learn to distinguish between relations and functions, work with composite functions, and understand inverse functions.

Learning Objectives

After completing this chapter, you will be able to:

• Distinguish between relations and functionsi
• Understand and use function notation
• Determine domain and range of functions
• Apply the vertical line test
• Find composite functions
• Determine and verify the existence of inverse functions
• Apply the horizontal line test
• Find inverse functions using algebraic methods

Key Concepts

1.1 Functions (Functions)

Understanding Relations vs Functions

A relation is any set of ordered pairs, while a function is a special type of relation where each element in the domain corresponds to exactly one element in the range.

Key Characteristics of Functions:

• Each input (object) maps to exactly one output (image)
• Multiple inputs can map to the same output
• No input can map to multiple outputs

Function Notation and Evaluation

The function notation $f(x)$ represents the value of function $f$ at input $x$. This can also be written as $f: x \to y$ where $y$ is the output.

Evaluation Examples:

• If $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$
• If $g(x) = x^2 - 4$, then $g(-2) = (-2)^2 - 4 = 0$

Vertical Line Test

A graph represents a function if any vertical line intersects the graph at most once.

Horizontal Line Test

A function is one-to-one if any horizontal line intersects the graph at most once.

Absolute Value Function

The absolute value function, $f(x) = |x|$, is a piecewise function defined as:

$$f(x) = |x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

Types of Functions

1. Discrete Function: Defined only at specific points
2. Continuous Function: Defined for all real numbers in an interval

Absolute Value Function

The absolute value function, f(x) = |x|, is a piecewise function defined as:

$$f(x) = |x| = \begin{cases} x, & \text{if } x \geq 0 \\ -x, & \text{if } x < 0 \end{cases}$$

1.2 Composite Functions

Definition

A composite function is formed by combining two or more functions, where the output of one function becomes the input of another.

Key Properties

• Order matters: $fg(x) \neq gf(x)$ in general
• Notation:
  • $fg(x) = f(g(x))$
  • $f^2(x) = f(f(x))$
  • $f^3(x) = f(f(f(x)))$

Composition Process

Composition Examples

Given $f(x) = 2x + 1$ and $g(x) = x^2 - 3$:

• $fg(x) = f(g(x)) = f(x^2 - 3) = 2(x^2 - 3) + 1 = 2x^2 - 5$
• $gf(x) = g(f(x)) = g(2x + 1) = (2x + 1)^2 - 3 = 4x^2 + 4x + 1 - 3 = 4x^2 + 4x - 2$

1.3 Inverse Functions

Definition

An inverse function, $f^{-1}(x)$, reverses the mapping of the original function. If $f$ maps $a$ to $b$, then $f^{-1}$ maps $b$ to $a$.

Conditions for Existence

A function has an inverse only if it is one-to-one:

• Each element in the domain maps to a unique element in the range
• The function passes the horizontal line test

Key Properties

• $ff^{-1}(x) = x$
• $f^{-1}f(x) = x$
• Domain of $f(x)$ = Range of $f^{-1}(x)$
• Range of $f(x)$ = Domain of $f^{-1}(x)$

Inverse Function Mapping

Finding Inverse Functions - Step-by-Step Method

Inverse Function Examples

Example 1: Linear Function Given $f(x) = 3x - 5$:

1. Let $y = 3x - 5$
2. Solve for $x$: $y + 5 = 3x$ ⇒ $x = \frac{y + 5}{3}$
3. Therefore, $f^{-1}(x) = \frac{x + 5}{3}$

Verification: $f(f^{-1}(x)) = 3\left(\frac{x + 5}{3}\right) - 5 = x + 5 - 5 = x$ ✓

Example 2: Quadratic Function with Restricted Domain Given $f(x) = x^2 + 4$ for $x \geq 0$:

1. Let $y = x^2 + 4$
2. Solve for $x$: $x^2 = y - 4$ ⇒ $x = \sqrt{y - 4}$ (since $x \geq 0$)
3. Therefore, $f^{-1}(x) = \sqrt{x - 4}$ for $x \geq 4$

Important Formulas and Methods

Function Notation

$$f(x) \text{ or } f: x \to y$$

Domain and Range

• Domain: All possible input values for which the function is defined
• Range: All possible output values produced by the function

Common Domains:

• $f(x) = \sqrt{x}$: Domain $x \geq 0$, Range $y \geq 0$
• $f(x) = \frac{1}{x}$: Domain $x \neq 0$, Range $y \neq 0$
• $f(x) = x^2$: Domain $x \in \mathbb{R}$, Range $y \geq 0$

Function Transformations

Key Function Families

Solved Examples

Example 1: Basic Function Operations

Given $f(x) = 2x + 3$ and $g(x) = x^2 - 1$, find:

• $f(4)$
• $g(-3)$
• $fg(2)$
• $gf(1)$

Solution:

1. $f(4) = 2(4) + 3 = 8 + 3 = 11$
2. $g(-3) = (-3)^2 - 1 = 9 - 1 = 8$
3. $fg(2) = f(g(2)) = f(2^2 - 1) = f(3) = 2(3) + 3 = 9$
4. $gf(1) = g(f(1)) = g(2(1) + 3) = g(5) = 5^2 - 1 = 24$

Example 2: Finding Inverse Function

Find the inverse of $f(x) = 3x - 5$.

Solution:

1. Let $y = 3x - 5$
2. Solve for $x$: $y + 5 = 3x$ ⇒ $x = \frac{y + 5}{3}$
3. Therefore, $f^{-1}(x) = \frac{x + 5}{3}$

Verification: $f(f^{-1}(x)) = 3\left(\frac{x + 5}{3}\right) - 5 = x + 5 - 5 = x$ ✓

Example 3: Domain and Range

Find the domain and range of $f(x) = \sqrt{x + 4}$.

Solution:

• Domain: $x + 4 \geq 0$ ⇒ $x \geq -4$
• Range: Since $\sqrt{x + 4} \geq 0$, range is $y \geq 0$

Example 4: Composite Function with Restricted Domain

Given $f(x) = \sqrt{x}$ and $g(x) = x^2 - 4$, find $fg(x)$ and its domain.

Solution:

1. $fg(x) = f(g(x)) = f(x^2 - 4) = \sqrt{x^2 - 4}$
2. Domain: $x^2 - 4 \geq 0$ ⇒ $x \leq -2$ or $x \geq 2$

Mathematical Proofs

Proof that $fg(x) \neq gf(x)$ in General

Counterexample: Let $f(x) = x + 1$ and $g(x) = 2x$

Then:

• $fg(x) = f(g(x)) = f(2x) = 2x + 1$
• $gf(x) = g(f(x)) = g(x + 1) = 2(x + 1) = 2x + 2$

Since $2x + 1 \neq 2x + 2$, $fg(x) \neq gf(x)$

Proof that Only One-to-One Functions Have Inverses

Theorem: A function $f$ has an inverse if and only if it is one-to-one.

Proof:

1. If $f$ has an inverse $f^{-1}$, then $f$ is one-to-one:

   • Assume $f(a) = f(b)$
   • Apply $f^{-1}$ to both sides: $f^{-1}(f(a)) = f^{-1}(f(b))$
   • Since $f^{-1}f(x) = x$, we get $a = b$
   • Therefore, $f$ is one-to-one

2. If $f$ is one-to-one, then $f$ has an inverse:

   • For each $y$ in the range of $f$, there exists exactly one $x$ such that $f(x) = y$
   • Define $f^{-1}(y) = x$ where $f(x) = y$
   • This is well-defined because $f$ is one-to-one

Real-World Applications

1. Population Growth Models

Composite functions can model multi-stage growth processes:

• Initial population: f(t) = $P_0$(1 + r)ᵗ
• Migration effect: g(P) = P + M
• Total model: gf(t) = $P_0$(1 + r)ᵗ + M

2. Economics

Revenue functions can be composed with demand functions:

• Price function: p(x) = 100 - 0.5x
• Revenue function: R(p) = p × x
• Composite: R(p(x)) = (100 - 0.5x)x = 100x - 0.5$x^2$

Complex Problem-Solving Techniques

Problem: Determine if f(x) = $x^2$ - 4x + 5 is one-to-one

Solution Strategy:

1. Check if the function passes the horizontal line test
2. Alternatively, check if the function is strictly increasing or decreasing
3. Or check if f(a) = f(b) implies a = b

Analysis: The function f(x) = $x^2$ - 4x + 5 is a parabola opening upwards with vertex at x = 2. Since it's not strictly increasing or decreasing over its entire domain, it's not one-to-one. However, it is one-to-one on the intervals (-∞, 2) and (2, ∞) separately.

Summary Points

• A function assigns exactly one output to each input
• Use the vertical line test to verify if a graph represents a function
• Composite functions combine multiple functions in sequence
• Inverse functions reverse the mapping of the original function
• Only one-to-one functions have inverses
• The horizontal line test determines if a function is one-to-one

Common Mistakes to Avoid

1. Assuming fg(x) = gf(x) - Order matters in composition
2. Forgiving domain restrictions - Always consider the domain when finding inverses
3. Misapplying the vertical line test - The test must work for ALL vertical lines
4. Ignoring piecewise functions - Absolute value functions are piecewise
5. Forgetting to verify inverses - Always check ff⁻¹(x) = x

SPM Exam Tips

Exam Strategies

1. Master function notation - Understand f(x) notation thoroughly
2. Practice composition - Work with multiple functions to understand composition order
3. Domain and range - Always specify domain and range when working with functions
4. Inverse verification - Verify your inverse functions by composition

Key Exam Topics

• Function definition and notation (20% of questions)
• Composite functions and their evaluation (30% of questions)
• Inverse functions and their conditions (25% of questions)
• Domain and range problems (15% of questions)
• Graph interpretation using line tests (10% of questions)

Time Management Tips

• Spend 3-4 minutes on basic function operations
• Allocate 5-7 minutes for composite function problems
• Reserve 6-8 minutes for inverse function questions
• Graph interpretation questions should take 4-5 minutes

Practice Problems

Level 1: Basic Functions

1. Given f(x) = 3x - 2, find f(5), f(-1), and f(0).
2. If g(x) = $x^2$ + 4x, calculate g(3) and g(-2).
3. Determine which of the following relations are functions:
   • {(1,2), (2,3), (3,4), (4,5)}
   • {(1,2), (1,3), (2,3), (2,4)}
   • {(x, y) | y = 2x + 1}

Level 2: Composite Functions

1. Given f(x) = 2x + 1 and g(x) = $x^2$ - 3, find:

   • fg(x)
   • gf(x)
   • $f^2$(x)
   • $g^2$(x)

2. If h(x) = 3x - 5 and k(x) = √x, find hk(16) and kh(1).

Level 3: Inverse Functions

1. Find the inverse of each function:

   • f(x) = 4x + 7
   • g(x) = 2 - 3x
   • h(x) = $x^2$ + 4 for x ≥ 0
   • k(x) = 1/x for x ≠ 0

2. Verify that f and f⁻¹ are indeed inverses by computing ff⁻¹(x) and f⁻¹f(x).

Did You Know? 📚

The concept of functions was formally developed in the 17th century by mathematicians like Leibniz and Bernoulli. The notation f(x) was introduced by Leonhard Euler in 1734, revolutionizing how we express mathematical relationships.

Quick Reference Guide

---

This chapter provides the foundation for all subsequent topics in Additional Mathematics. Master these concepts thoroughly as they will be used extensively in chapters on quadratic functions, calculus, and beyond.