Functions and Graphs: Fundamental Concepts and Applications

Section 1.2: Functions and Graphs

Objectives

This section introduces the foundational concepts of functions and relations, their graphical representations, and how to determine domains and ranges. Students will learn to:

• Determine whether a correspondence or relation is a function.

• Find function values using a formula.

• Graph functions.

• Identify whether a graph represents a function.

• Find the domain and range of a function.

• Solve applied problems using functions.

Functions

A function is a specific type of correspondence between two sets: the domain (inputs) and the range (outputs). Each member of the domain is paired with exactly one member of the range.

• Definition: A function is a rule that assigns to each element in the domain exactly one element in the range.

• Domain: The set of all possible input values (often represented by x).

• Range: The set of all possible output values (often represented by y or f(x)).

• Not every correspondence between two sets is a function; the key requirement is that each input has only one output.

Example

Consider the following correspondence:

• $-6 ightarrow 36$

• $6 ightarrow 36$

• $-3 ightarrow 9$

• $0 ightarrow 0$

This is a function because each member of the domain corresponds to exactly one member of the range. Multiple domain elements can share the same range value.

Relations

A relation is a general correspondence between two sets, the domain and the range. Unlike functions, a relation does not require each input to have only one output.

• Definition: A relation pairs elements of the domain with elements of the range, but a single input may correspond to multiple outputs.

Example

Consider the relation: $\{(9, -5), (9, 5), (2, 4)\}$

• This is not a function because the input $9$is paired with both$-5$and $5$.

• Domain: $\{9, 2\}$

• Range: $\{-5, 5, 4\}$

Example (Function)

Consider the relation: $\{(-2, 5), (5, 7), (0, 1), (4, -2)\}$

• This is a function because no input is paired with more than one output.

• Domain: $\{-2, 5, 0, 4\}$

• Range: $\{5, 7, 1, -2\}$

Example (Function with Same Output)

Consider the relation: $\{(-5, 3), (0, 3), (6, 3)\}$

• This is a function because each input is paired with only one output, even though multiple inputs share the same output.

• Domain: $\{-5, 0, 6\}$

• Range: $\{3\}$

Function Notation

Functions are commonly written in the form $f(x)$, where:

• Inputs: Values substituted for $x$ (members of the domain).

• Outputs: Values of $f(x)$ (members of the range).

• $f(x)$ is read as "f of x," "f at x," or "the value of f at x."

Example

Given $f(x) = 2x^2 - x + 3$, find $f(0)$ and $f(-7)$:

• $f(0) = 2(0)^2 - 0 + 3 = 0 - 0 + 3 = 3$

• $f(-7) = 2(-7)^2 - (-7) + 3 = 2 imes 49 + 7 + 3 = 98 + 7 + 3 = 108$

Graphs of Functions

Functions can be represented graphically by plotting ordered pairs $(x, f(x))$ on a coordinate plane.

• Each point on the graph corresponds to an input-output pair.

• Graphing helps visualize the behavior of the function.

Example

Graph $f(x) = x^2 - 5$ by creating a table of values and plotting the points.

Vertical-Line Test

The vertical-line test is a graphical method to determine if a graph represents a function.

• If any vertical line crosses the graph more than once, the graph does not represent a function.

• If every vertical line crosses the graph at most once, the graph does represent a function.

Finding Domains of Functions

The domain of a function is the set of all input values for which the function is defined.

• For functions given by a formula, the domain consists of all real numbers for which the formula yields a real output.

• Inputs that result in division by zero or taking the square root of a negative number are excluded from the domain.

Example

Find the domain of $f(x) = \frac{1}{x-3}$:

• $f(1) = \frac{1}{1-3} = \frac{1}{-2}$ is defined, so $1$ is in the domain.

• $f(3) = \frac{1}{3-3} = \frac{1}{0}$ is undefined, so $3$ is not in the domain.

Example

Find the domain of $h(x) = \frac{3x+7}{x^2+2x-3}$:

• Set denominator not equal to zero: $x^2 + 2x - 3 = 0$

• Factor: $(x+3)(x-1) = 0$

• So $x = -3$ and $x = 1$ are excluded from the domain.

• Domain: All real numbers except $x = -3$ and $x = 1$.

Visualizing Domain and Range

On a graph:

• Domain: Set of inputs, found on the horizontal axis.

• Range: Set of outputs, found on the vertical axis.

Example

For $f(x) = \sqrt{x+4}$:

• Domain: $x \geq -4$ (since the expression under the square root must be non-negative)

• Range: $f(x) \geq 0$

Summary Table: Function vs. Relation

Feature	Function	Relation
Definition	Each input has exactly one output	Inputs may have multiple outputs
Vertical-Line Test	Passes	Fails if any vertical line crosses more than once
Notation	$f(x)$	Set of ordered pairs
Domain	All valid inputs	All first elements of ordered pairs
Range	All possible outputs	All second elements of ordered pairs

Additional info: These notes expand on the provided slides and text, adding definitions, examples, and a summary table for clarity and completeness.