# Functions

## Overview

Welcome to the fascinating world of Functions in Further Mathematics. Functions play a crucial role in mathematics, serving as essential tools for modeling relationships between variables and analyzing various phenomena.

Understanding the Notation of Functions: In the realm of functions, notation is key to expressing relationships between inputs and outputs. For instance, a function f can be defined as f : x → 3x+4, where x belongs to the set of real numbers. This notation signifies that the function f maps each input x to the output 3x+4.

Determining Range and Image: The range of a function refers to the set of all possible output values it can attain, while the image is the actual output set for a given domain. By understanding these concepts, we can gain insights into the behavior and limits of functions.

Finding Inverse Functions: One-to-one functions hold a special property where each input corresponds to a unique output. Determining the inverse function involves swapping the roles of inputs and outputs. For example, if f is f: x → √x, then the inverse relation f-1: x → x^2 can be obtained.

Exploring Composite Functions: The composition of functions, denoted as fog(x) = f(g(x)), allows us to combine multiple functions to create new relationships. This concept is invaluable in analyzing complex mathematical scenarios and problem-solving.

Identifying Function Properties: Functions exhibit various properties such as closure, commutativity, associativity, and distributivity, which govern their behavior under different operations. Understanding these properties aids in manipulating functions effectively.

Graphical Representation of Functions: Visualizing functions through graphs provides a clear depiction of their behavior and characteristics. Graphs help us comprehend the trends, domain, range, and critical points of functions, facilitating a deeper understanding.

Logic and Set Theory: In addition to functions, this course material delves into set theory, including concepts like disjoint sets, Venn diagrams, and the use of sets to solve problems. Understanding the syntax of true or false statements, logic rules, and implications is crucial in mathematical reasoning.

This course material will equip you with the foundational knowledge and skills needed to navigate the intricate world of functions, sets, and logic in Further Mathematics. Through engaging explanations, illustrative examples, and interactive learning tasks, you will master the art of analyzing relationships, solving complex problems, and advancing your mathematical prowess.

## Objectives

1. Apply the concept of composite functions
2. Explore the graphical representation of functions
3. Find the inverse of one-to-one functions
4. Solve problems using functions and their inverses
5. Determine the range and image of functions
6. Understand the notation of functions
7. Identify the properties of functions

## Lesson Note

In mathematics, a function is a relationship or expression involving one or more variables. Functions describe how one quantity depends on another, and they are fundamental building blocks in mathematics. This article will delve into different aspects of functions, including composite functions, inverse functions, graphical representations, and more.

## Lesson Evaluation

Congratulations on completing the lesson on Functions. Now that youve explored the key concepts and ideas, its time to put your knowledge to the test. This section offers a variety of practice questions designed to reinforce your understanding and help you gauge your grasp of the material.

You will encounter a mix of question types, including multiple-choice questions, short answer questions, and essay questions. Each question is thoughtfully crafted to assess different aspects of your knowledge and critical thinking skills.

Use this evaluation section as an opportunity to reinforce your understanding of the topic and to identify any areas where you may need additional study. Don't be discouraged by any challenges you encounter; instead, view them as opportunities for growth and improvement.

1. Find the inverse of the function f(x) = 2x - 5. A. f-1(x) = x/2 + 5/2 B. f-1(x) = 2x + 5 C. f-1(x) = (x + 5)/2 D. f-1(x) = 2x - 5 Answer: A. f-1(x) = x/2 + 5/2
2. Given f(x) = 3x + 2 and g(x) = x^2, find f(g(x)). A. 3x^2 + 2 B. 3x^2 + 6 C. x^2 + 2 D. x^2 + 3x + 2 Answer: B. 3x^2 + 6
3. If f(x) = √(4x - 1), what is the range of the function f(x)? A. (-∞, 1] B. [0, ∞) C. (-∞, ∞) D. [1, ∞) Answer: D. [1, ∞)
4. Find the image of the function h(x) = x^2 - 9. A. {y ∈ R: y ≥ -9} B. {y ∈ R: y ≥ 0} C. {y ∈ R: y ≤ -9} D. {y ∈ R: y ≤ 0} Answer: A. {y ∈ R: y ≥ -9}
5. If f(x) = 4x + 3 and g(x) = x^2 - 5, determine f o g(x). A. 4x^2 - 17 B. 4x^2 - 17x - 15 C. 4x^2 - 12 D. 4x^2 + 3 Answer: A. 4x^2 - 17
6. For the function j(x) = √(x + 7), identify the domain of the function j(x). A. {x ∈ R: x ≥ -7} B. {x ∈ R: x > -7} C. {x ∈ R: x ≤ -7} D. {x ∈ R: x ≥ 7} Answer: A. {x ∈ R: x ≥ -7}
7. If f(x) = 5x - 2 and f is a one-to-one function, what is the inverse function? A. f-1(x) = (x + 2)/5 B. f-1(x) = -5x + 2 C. f-1(x) = (x - 2)/5 D. f-1(x) = 5x - 2 Answer: C. f-1(x) = (x - 2)/5
8. What is the composite function for f(x) = x^2 - 4 and g(x) = 2x + 3? A. 2(x^2) - 8 B. 2x^2 - 8 C. x^2 + 3 D. x^2 - 4 Answer: B. 2x^2 - 8
9. If f(x) = √x and g(x) = x - 1, determine the composite function f o g(x). A. √(x - 1) B. √(x + 1) C. (x - 1)^(1/2) D. (x + 1)^(1/2) Answer: A. √(x - 1)

## Past Questions

Wondering what past questions for this topic looks like? Here are a number of questions about Functions from previous years

Question 1

Two functions f and g are defined on the set of real numbers, R, by

f:x → x2  + 2 and g:x → 1x+2.Find the domain of (g∘f)1

Practice a number of Functions past questions