Precalculus Study Notes: Functions and Their Graphs

Section 1.2: Functions and Graphs

Objectives

• Determine whether a correspondence or a relation is a function.

• Find function values (outputs) using a formula or a graph.

• Graph functions.

• Determine whether a graph is that of a function.

• Find the domain and range of a function.

• Solve applied problems using functions.

Functions and Relations

Definition of a Function

A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.

• Not every correspondence between two sets is a function.

• Each input (domain) has only one output (range), but different inputs can have the same output.

Definition of a Relation

A relation is a correspondence between the domain and the range such that each member of the domain corresponds to at least one member of the range.

• All functions are relations, but not all relations are functions.

Examples: Determining Functions and Relations

• Example 1: A mapping where each domain element points to exactly one range element is a function.

• Example 2: Consider the set of ordered pairs:

  • {(9, -5), (9, 5), (2, 4)}: Not a function because 9 is paired with both -5 and 5.

  • Domain: {9, 2}; Range: {-5, 5, 4}

  • {(-2, 5), (5, 7), (0, 1), (4, -2)}: Is a function because no first coordinate repeats.

  • Domain: {-2, 5, 0, 4}; Range: {5, 7, 1, -2}

  • {(-5, 3), (0, 3), (6, 3)}: Is a function because no first coordinate repeats.

  • Domain: {-5, 0, 6}; Range: {3}

Function Notation

Inputs and Outputs

• The input (domain) is usually denoted by x.

• The output (range) is the value of the function at x, denoted f(x).

• Read as “f of x,” “f at x,” or “the value of f at x.”

Examples: Evaluating Functions

• Given $f(x) = 2x^2 - x + 3$:

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

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

• $f(5a) = 2(5a)^2 - 5a + 3 = 2 \cdot 25a^2 - 5a + 3 = 50a^2 - 5a + 3$

• $f(a-4) = 2(a-4)^2 - (a-4) + 3 = 2(a^2 - 8a + 16) - a + 4 + 3 = 2a^2 - 16a + 32 - a + 4 + 3 = 2a^2 - 17a + 39$

Graphing Functions

Plotting Functions

• To graph a function, find ordered pairs $(x, f(x))$ and plot them on the coordinate plane.

• Connect the points smoothly if the function is continuous.

Examples: Graphing

• For $f(x) = x^2 - 5$, create a table of values for $x$ and plot the corresponding $f(x)$ values.

• For $f(x) = x^2 - x$, plot points for several $x$ values and connect them.

• For $f(x) = \sqrt{x+4}$, plot points for $x \geq -4$ (since the square root is defined for non-negative arguments).

Finding Function Values from Graphs

• To find $f(a)$ from a graph, locate $x = a$ on the horizontal axis, move vertically to the graph, then horizontally to the vertical axis to read $f(a)$.

• Example: For $f(x) = x^2 - 6$, to find $f(-3)$, locate $x = -3$, move up to the graph, then over to the $y$-axis to read the value.

The Vertical Line Test

Determining if a Graph Represents a Function

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

• This is called the vertical line test.

• Example: Some graphs pass the test (are functions), others do not.

Domain and Range of Functions

Finding the Domain

• The domain is the set of all input values ($x$) for which the function is defined as a real number.

• Exclude values that make the denominator zero or result in an undefined expression (such as a negative under a square root).

Examples: Determining Domain

• Given $f(x) = \frac{1}{x-3}$:

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

• $f(3)$ is undefined (division by zero), so 3 is not in the domain.

• Given $h(x) = \frac{3x - x + 7}{x^2 + 2x - 3}$:

• Set denominator to zero: $x^2 + 2x - 3 = 0 \implies (x+3)(x-1) = 0 \implies x = -3$ or $x = 1$

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

Visualizing Domain and Range

• The domain is found on the horizontal axis of the graph.

• The range is found on the vertical axis.

• Example: For $f(x) = \sqrt{x+4}$, the domain is $[-4, \infty)$ and the range is $[0, \infty)$.

Summary Table: Function vs. Relation

Concept	Definition	Example
Function	Each input has exactly one output	{(1,2), (2,3), (3,4)}
Relation	Each input has at least one output	{(1,2), (1,3), (2,4)}

Key Formulas and Notation

• Function notation: $f(x)$

• Quadratic function: $f(x) = ax^2 + bx + c$

• Square root function: $f(x) = \sqrt{x}$

• Rational function: $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$