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 defined by , the domain is , the codomain is , and the range is .

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 passes the VLT (is a function), but the graph of (a circle) does not.

Basic Classes of Functions

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

• Polynomial:

• Rational:

• Radical:

• Exponential: or

• Logarithmic: or

• Periodic: , , etc.

• Piecewise:

• Composite:

Characteristics of Functions

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

Even and Odd Functions

• Even: is even if for all in the domain.

• Odd: is odd if for all in the domain.

Examples:

• is even.

• is odd.

• is odd; is even.

Increasing and Decreasing Functions

• is increasing on an interval if whenever .

• is strictly increasing if for all .

• is decreasing if whenever .

• is strictly decreasing if for all .

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., , , ).

Examples:

• is algebraic.

• 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:

• Domain: ,

• Range: , but precise range may require graphing or calculus tools.

• Example 2:

• Domain: Solve and (always true for denominator).

• Domain: or

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:

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 is a classic example of a piecewise function:

Properties of Absolute Value:

Solving Absolute Value Inequalities

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

• Example: Solve

• Case 1:

• Case 2:

• Solution:

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.