Review and Basic Classes of Functions – MAT135 Differential Calculus (University of Toronto Mississauga)

Chapter 1 – Functions and Graphs

Introduction to Functions

Functions are foundational objects in calculus, providing a way to model relationships between quantities. Understanding their properties and representations is essential for further study in calculus.

• Definition: A function consists of a set of inputs (the domain), a set of outputs (the range), and a rule for assigning each input to exactly one output.

• Domain: The set of all possible inputs for the function.

• Codomain: The set of potential outputs (not always equal to the range).

• Range: The set of actual outputs produced by the function.

Example: For $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$, the domain is $\mathbb{R}$, the codomain is $\mathbb{R}$, and the range is $[0, \infty)$.

Additional info: The distinction between codomain and range is important for understanding surjective (onto) functions.

Ways to Represent a Function

There are four main ways to represent functions. Mastery of these forms and the ability to switch between them is crucial for problem-solving in calculus.

1. A table of values

2. A graph

3. A formula

4. A description (verbal or contextual)

Why Study Functions?

Functions allow us to mathematically model real-world phenomena, enabling analysis, prediction, and understanding of complex systems.

• Mathematical models use functions to describe relationships between variables.

• Modeling process: Define → Translate → Analyze → Interpret.

The Vertical Line Test (VLT)

The Vertical Line Test is a graphical method to determine if a curve represents a function.

• VLT: If any vertical line intersects a set of points (a curve) more than once, the set does not represent a function.

Example: The graph of $y = x^2$ passes the VLT (is a function), but the graph of $x^2 + y^2 = 1$ (a circle) does not.

Basic Classes of Functions

Functions can be classified into several basic types, each with unique properties and applications.

• Polynomial: $f(x) = a_n x^n + \ldots + a_1 x + a_0$

• Rational: $f(x) = \frac{p(x)}{q(x)}$

• Radical: $f(x) = \sqrt[n]{x}$

• Exponential: $f(x) = a^{x}$ or $f(x) = e^{x}$

• Logarithmic: $f(x) = \log_a(x)$ or $f(x) = \ln(x)$

• Periodic: $f(x) = \sin(x)$, $f(x) = \cos(x)$, etc.

• Piecewise: $f(x) = \begin{cases} f_1(x), & x < a \\ f_2(x), & x \geq a \end{cases}$

• Composite: $f(g(x))$

Characteristics of Functions

Understanding the behavior of functions involves examining their symmetry, monotonicity, and algebraic structure.

Even and Odd Functions

• Even: $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in the domain.

• Odd: $f(x)$ is odd if $f(-x) = -f(x)$ for all $x$ in the domain.

Examples:

• $f(x) = x^2$ is even.

• $f(x) = x^3$ is odd.

• $f(x) = \sin(x)$ is odd; $f(x) = \cos(x)$ is even.

Increasing and Decreasing Functions

• $f(x)$ is increasing on an interval if $f(x_1) < f(x_2)$ whenever $x_1 < x_2$.

• $f(x)$ is strictly increasing if $f(x_1) < f(x_2)$ for all $x_1 < x_2$.

• $f(x)$ is decreasing if $f(x_1) > f(x_2)$ whenever $x_1 < x_2$.

• $f(x)$ is strictly decreasing if $f(x_1) > f(x_2)$ for all $x_1 < x_2$.

Additional info: Determining monotonicity is easier with calculus tools (derivatives).

Algebraic and Transcendental Functions

• Algebraic: A function is algebraic if it can be constructed from a finite number of algebraic operations (addition, subtraction, multiplication, division, and taking roots).

• Transcendental: A function is transcendental if it is not algebraic (e.g., $e^x$, $\sin(x)$, $\log(x)$).

Examples:

• $f(x) = 2x + 7$ is algebraic.

• $f(x) = e^x$ is transcendental.

Domain and Range: Examples and Strategies

Finding the domain and range of a function is a key skill in calculus. Restrictions often arise from denominators and even roots.

• Example 1: $f(x) = \frac{\sqrt{x-5}}{x}$

• Domain: $x \geq 5$, $x \neq 0$

• Range: $y \geq 0$, but precise range may require graphing or calculus tools.

• Example 2: $f(x) = \frac{\sqrt{x^2-4x}}{x^2+5}$

• Domain: Solve $x^2-4x \geq 0$ and $x^2+5 \neq 0$ (always true for denominator).

• Domain: $x \leq 0$ or $x \geq 4$

Additional info: Graphs are often used to estimate the range when algebraic methods are insufficient.

Piecewise Functions

A piecewise function is defined by different formulas on different parts of its domain.

• Example: $f(x) = \begin{cases} x^2, & x < 1 \\ 2-x, & 1 \leq x < 3 \\ x-1, & x \geq 3 \end{cases}$

To determine if a piecewise function is a function, check that each input corresponds to exactly one output.

The Absolute Value Function

The absolute value function $f(x) = |x|$ is a classic example of a piecewise function:

• $|x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$

Properties of Absolute Value:

Inequality	Equivalent Form	Graph
$|x| = a$	$x = -a$ or $x = a$	Two points at $-a$ and $a$
$|x| < a$	$-a < x < a$	Interval between $-a$ and $a$
$|x| \leq a$	$-a \leq x \leq a$	Closed interval $[-a, a]$
$|x| > a$	$x < -a$ or $x > a$	Two rays outside $-a$ and $a$
$|x| \geq a$	$x \leq -a$ or $x \geq a$	Two rays including $-a$ and $a$

Solving Absolute Value Inequalities

To solve inequalities involving absolute value, consider both the positive and negative cases.

• Example: Solve $|2x + 6| \geq 8$

• Case 1: $2x + 6 \geq 8 \implies x \geq 1$

• Case 2: $2x + 6 \leq -8 \implies x \leq -7$

• Solution: $x \in (-\infty, -7] \cup [1, \infty)$

Graphical interpretation helps visualize the solution set.

Additional info: These foundational concepts are essential for understanding limits, continuity, derivatives, and integrals in subsequent calculus chapters.