Functions and Their Graphs: Precalculus Study Notes (Section 1.1)

Functions and Their Graphs

Section 1.1: 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

• Determine the domain of a function defined by an equation

• Find the sum, difference, product, and quotient of functions

Relations

Describing a Relation

A relation is a correspondence between two sets, where each element from the first set is paired with one or more elements from the second set. Relations are often used to describe how one variable depends on another.

• Ordered pairs are used to represent relations: (x, y), where x is from the first set (domain) and y is from the second set (range).

• We say "x corresponds to y" or "y depends on x" and write this as x → y.

Example:

Consider the relation between states and the number of representatives each state has in the House of Representatives:

State	Number of Representatives
Mississippi	4
Louisiana	6
Alabama	7
Georgia	14
Florida	27

• The domain is {Mississippi, Louisiana, Alabama, Georgia, Florida}.

• The range is {4, 6, 7, 14, 27}.

Functions

Definition of a Function

A function from set X into set Y is a relation that pairs each element of X with exactly one element of Y.

• Usually, X and Y are sets of real numbers.

• Functions can also be defined for complex numbers.

• Notation: If f is a function, we write $y = f(x)$, where x is the independent variable and y is the dependent variable.

Example:

Given the following mapping:

Store	Price ($)
Maziply	9.99
Amazon	10.80
Primegood2	10.99
Toys & Co.	11.83
kids create fun	14.99
GoodLocker	14.99

• The domain is {Maziply, Amazon, Primegood2, Toys & Co., kids create fun, GoodLocker}.

• The range is {9.99, 10.80, 10.99, 11.83, 14.99}.

• This is a function because each store corresponds to exactly one price.

Determining Whether a Relation is a Function

To determine if a relation is a function, check if each element in the domain is paired with only one element in the range.

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

Example:

Consider the set of ordered pairs: {(1, 5), (3, 9), (5, 1), (9, 2)}

• Domain: {1, 3, 5, 9}

• Range: {5, 9, 1, 2}

• This is a function because no domain element is repeated with a different range value.

Counterexample:

Consider: {(5, 1), (5, -3)}

• Domain: {5}

• Range: {1, -3}

• This is not a function because 5 is paired with two different values.

Functions Defined by Equations

Many functions are defined by equations. To determine if an equation defines y as a function of x, solve for y and check if each x yields only one y.

Example:

• $y = 3x + 7$ is a function because each x gives exactly one y.

• $6x + 3y^2 = 2$ is not a function because some x values yield two y values.

Function Notation and Evaluation

Function Notation

Functions are often written as $f(x)$, where:

• x is the independent variable (input).

• f(x) is the dependent variable (output).

Example:

Given $f(x) = 2x^2 - 5x$, find $f(3)$:

• $f(3) = 2(3)^2 - 5(3) = 18 - 15 = 3$

Explicit and Implicit Form

• Explicit form: y is written directly in terms of x, e.g., $y = f(x) = -2x + 4$

• Implicit form: y and x are mixed, e.g., $y + 2x = 4$

Difference Quotient

Definition

The difference quotient of a function f at x is:

• $rac{f(x+h) - f(x)}{h}$

Example:

For $f(x) = rac{1}{x+1}$, the difference quotient is:

• $rac{f(x+h) - f(x)}{h} = rac{ rac{1}{x+h+1} - rac{1}{x+1}}{h}$

• Simplify numerator: $rac{1}{x+h+1} - rac{1}{x+1} = rac{(x+1) - (x+h+1)}{(x+h+1)(x+1)} = rac{-h}{(x+h+1)(x+1)}$

• So, $rac{-h}{h(x+h+1)(x+1)} = rac{-1}{(x+h+1)(x+1)}$

Domain of a Function

Finding the Domain

The domain of a function is the set of all permissible input values (x) for which the function is defined.

• If the function has a denominator, exclude values that make the denominator zero.

• If the function has an even-index radical, exclude values that make the radicand negative.

Example:

Find the domain of $F(x) = rac{ oot{2}{2x+6}}{x-2}$

• Require $2x+6 ext{ to be } ext{non-negative} ightarrow 2x+6 ext{ } extgreater{} 0 ightarrow x extgreater{} -3$

• Exclude $x=2$ (denominator zero)

• Domain: $oxed{ ext{All } x ext{ such that } x extgreater{} -3, x eq 2}$

Application Example:

Volume of a cube as a function of side length $s$:

• $V(s) = s^3$

• Domain: $s > 0$ (side length must be positive)

Operations on Functions

Sum, Difference, Product, and Quotient

Given functions $f$ and $g$:

• Sum: $(f+g)(x) = f(x) + g(x)$

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

• Product: $(fg)(x) = f(x)g(x)$

• Quotient: $igg( rac{f}{g}igg)(x) = rac{f(x)}{g(x)}$, $g(x) eq 0$

Example:

If $f(x) = rac{x-4}{x}$ and $g(x) = rac{2x-4}{x+3}$, then:

• $(f+g)(x) = rac{x-4}{x} + rac{2x-4}{x+3}$

• Domain: All $x$ except $x=0$ and $x=-3$

Summary Table: Relation vs. Function

Type	Definition	Example
Relation	Any pairing between elements of two sets	(Mississippi, 4), (Louisiana, 6)
Function	Each domain element paired with exactly one range element	$y = 3x + 7$

Additional info: Some examples and explanations have been expanded for clarity and completeness.