Section 2.1 Functions

Objectives

• Describe a relation

• Determine whether a relation represents a function

• Use function notation and find the value of a function

• Find the difference quotient of a function

• Find the domain of a function defined by an equation

• Form the sum, difference, product, and quotient of two functions

Relations

Definition and Representation

A relation is a correspondence between two sets: a set X (the domain) and a set Y (the range). In a relation, each element from the domain corresponds to at least one element from the range. The notation x → y indicates that y depends on x, where x is the input and y is the output.

Relations can be expressed in several ways:

• Verbally (describing the relationship in words)

• Numerically (using tables or sets of ordered pairs)

• Graphically (plotting points on a coordinate plane)

• As a mapping (drawing arrows from domain elements to range elements)

Example: Describing a Relation

Consider the average retail price of gasoline in the U.S. over several years:

• Domain: {2010, 2012, 2014, 2016, 2018}

• Range: {$2.78, $3.62, $3.36, $2.14, $2.72}

• Ordered pairs: {(2010, $2.78), (2012, $3.62), (2014, $3.36), (2016, $2.14), (2018, $2.72)}

Functions

Definition and Properties

A function from set X into set Y is a relation that associates with each element of X exactly one element of Y. The set X is called the domain, and the set of all images (outputs) is called the range. The range may be a proper subset of Y.

Examples: Determining Functions from Mappings

• Example a: Menu items and their prices. Each item has exactly one price, so this is a function.

• Example b: Friends and their step counts. Each friend has exactly one step count, so this is a function.

• Example c: Students and college admittances. If a student is admitted to more than one college, the relation is not a function.

Examples: Ordered Pairs and Functions

• If no two ordered pairs have the same first element with different second elements, the relation is a function.

• If any first element is paired with more than one second element, the relation is not a function.

Examples: Equations as Functions

• The equation $y = 3x + 7$ defines a function because each input $x$ yields exactly one output $y$.

• Equations that yield more than one output for a single input (e.g., $x^2 + y^2 = 1$) do not define functions.

Function Notation and Evaluation

Function Notation

If $f$ is a function, then for each $x$ in the domain, $f(x)$ denotes the value of the function at $x$. The variable $x$ is called the independent variable (or argument), and $y = f(x)$ is the dependent variable.

Evaluating Functions

• To evaluate $f(a)$, substitute $a$ for $x$ in the function's formula.

• Examples include substituting values, expressions, or variables for $x$.

Calculator Examples

• Graphing and evaluating functions using a calculator can verify results.

Implicit and Explicit Forms of Functions

A function is given implicitly if defined by an equation involving $x$ and $y$. If it is possible to solve for $y$ in terms of $x$, the function is given explicitly as $y = f(x)$.

Difference Quotient

Definition

The difference quotient of a function $f$ at $x$ is given by:

$ \frac{f(x+h) - f(x)}{h} $

This expression is foundational for calculus, representing the average rate of change of the function over an interval of length $h$.

Domain of a Function

Finding the Domain

• Start with all real numbers.

• Exclude values that make denominators zero.

• Exclude values that make even-indexed radicals negative.

Express the domain using interval notation, set notation, or words as appropriate.

Examples: Finding the Domain

• For $f(x) = x^2 + 1$, the domain is all real numbers.

• For $g(x) = \frac{x}{x^2 - 25}$, exclude $x = 5$ and $x = -5$.

• For $h(t) = \sqrt{5 - 5t}$, require $5 - 5t \geq 0$.

• For $F(x) = \frac{\sqrt{2x + 6}}{x - 2}$, require $2x + 6 \geq 0$ and $x \neq 2$.

Application Example: Volume of a Cube

The volume $V$ of a cube with side length $s$ is given by:

$ V = s^3 $

The domain is $s > 0$ (since side length must be positive).

Operations on Functions

Definitions

• Sum: $(f + g)(x) = f(x) + g(x)$

• Difference: $(f - g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x)g(x)$

• Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

The domain of each operation is the intersection of the domains of $f$ and $g$, with additional restrictions for division (where $g(x) \neq 0$).

Summary Table: Function Operations and Domains