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Precalculus Learning Guide Overview

This learning guide is designed for Bachelor of Engineering Mathematics (EMA105B) students at Tshwane University of Technology. Compiled by Mrs. Djankou Priscille and Dr. Aphane, with contributions from tutors, this document provides various problems and tutorials to aid learning. It is important to note that this guide supplements but does not replace the prescribed textbook. Students are encouraged to purchase the specific textbooks for comprehensive study. The guide is organized by chapters, each offering problems sorted by topics, with some solutions mentioned at the end.

Chapter Structure

• Chapter 1: Number System

• Chapter 2: Functions

• Chapter 3: Types of Functions

• Chapter 4: Inequalities

• Chapter 5: Logarithmic and Exponential Functions

• Chapter 6: Continuous Functions

• Chapter 7: Complex Numbers

• Chapter 8: Function Transformations

• Chapter 9: Vectors

• Chapter 10: Matrices

• Chapter 11: Linear Dependence

Chapter 1: Number System

Number System Definitions

• The set of real numbers oldsymbol{ ext{ℝ}} consists of:

  • Rational Numbers oldsymbol{ ext{ℚ}}: can be expressed as $rac{a}{b}$ where $a$ and $b$ are integers and $b <br>eq 0$. Example: 6, -3/4, and 1.75.

  • Irrational Numbers: cannot be expressed as a fraction. Example: $ext{π}, ext{√2}$.

• Important Notations:

  • oldsymbol{ ext{ℤ}} : Set of integers

  • oldsymbol{ ext{ℕ}} : Set of natural numbers

  • oldsymbol{ ext{ℎ}} : Set of whole numbers

Key Examples

• Rational Numbers: 6, -2, 1/2

• Irrational Numbers: π, 0.101110…

Chapter 2: Functions

Definition of a Function

A function is a binary relation from set A (domain) to set B (range) that assigns every element in A exactly one element in B.

• Domain: Set of input values for a function.

• Range: Set of output values for a function.

Types of Functions

1. Injective (One-to-One): Each output is assigned to only one input.

2. Surjective (Onto): Every element in the range corresponds to an element in the domain.

3. Bijective: A function that is both injective and surjective.

Example Functions

1. Find the domain and range of $f(x) = ext{√}(x - 5)$; Domain: $ext{ℝ} | x ext{≥} 5$, Range: $y ext{≥} 0$

Chapter 3: Types of Functions

Polynomial Functions

• General form: $P(x) = a<em>nx^n + a</em>{n-1}x^{n-1} + ext{…} + a<em>1x + a</em>0$; with $a_n$ being the leading coefficient.

Long Division of Polynomials

Presenting the steps involved in dividing polynomials.

Chapter 4: Inequalities

Definitions and Rules

An inequality describes a relation between two expressions that are not equal. Various properties describe how to manipulate and solve inequalities involving absolute values. Examples provided in context.

Chapter 5: Logarithmic and Exponential Functions

Key Concepts

• Logarithms: Relation between the base of a number and its exponent.

• Exponential Functions: Functions of the form $f(x) = ab^x$ where b is a constant.

Exercises

• Convert decimals to fractions, solve logarithmic equations, evaluate exponential growth models.

Chapter 6: Continuous and Piecewise Continuous Functions

Definitions

A piecewise function is continuous in intervals but may have discontinuities at certain points. Illustrative examples provided.

Chapter 7: Complex Numbers

Basic Terminology

Complex numbers consist of a real part and an imaginary part, expressed in standard form as $z = x + iy$ where $x$ = Re(z) and $y$ = Im(z).

Operations on Complex Numbers

Addition, subtraction, multiplication, and division of complex numbers are detailed.

Chapter 8: Transformations of Functions

Types of Transformations

Vertical and horizontal shifts, reflections, stretches, and compressions are discussed with graphical illustrations.

Chapter 9: Vectors

Vectors Overview

Difference between scalar and vector quantities; operations on vectors including addition and subtraction, dot product, cross product.

Chapter 10: Matrices

Introduction to Matrices

Understanding matrices, their properties, and operations such as addition, subtraction and multiplicative inverses are essential for solving equations and transformations.

Chapter 11: Linear Dependence and Independence

Definitions

A set of vectors is linearly independent if the equation $c<em>1v</em>1 + c<em>2v</em>2 + … + c<em>nv</em>n = 0$ has only the trivial solution. Examples of verifying linear dependence or independence are illustrated.

Completing the Guide

Tutorial Exercises

Problem sets assist students in applying theoretical knowledge to practical problems, ensuring understanding across discussed topics.

The guide trusts that students will find it beneficial and are encouraged to reach out for any clarity or improvements. This document serves as a foundational resource to excel in Engineering Mathematics topics at Tshwane University.