Study Guide: Basics of Functions and Their Graphs (Precalculus)

Functions and Relations

Definitions and Properties

A relation is any set of ordered pairs. The domain of a relation is the set of all first components (inputs), and the range is the set of all second components (outputs). A function is a special type of relation where each element in the domain corresponds to exactly one element in the range.

• Domain: All possible input values (x-values).

• Range: All possible output values (y-values).

• Function: Each input has only one output.

Example

Given the relation: (0, 187), (1, 212), (2, 225), (2, 238), the domain is {0, 1, 2} and the range is {187, 212, 225, 238}. Since the input 2 corresponds to two outputs (225 and 238), this relation is not a function.

Determining Functions from Graphs

The Vertical Line Test

The Vertical Line Test is used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not define y as a function of x.

• If a vertical line crosses the graph more than once, it is not a function.

• If every vertical line crosses the graph at most once, it is a function.

This graph fails the vertical line test and is not a function.

This graph passes the vertical line test and is a function.

Determining Functions from Equations

Solving for y

If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define y as a function of x.

• Example: $x^2 + y^2 = 1$ gives two values of y for most x, so it is not a function.

Function Notation

Understanding f(x)

Function notation uses f(x) to represent the value of the function at x. For example, if $f(x) = x^2 - 2x + 7$, then $f(-5)$ means substitute -5 for x.

• $f(x)$ is read as "f of x".

• Represents the output for a given input x.

Graphing Functions

Visual Representation

The graph of a function is the set of its ordered pairs plotted in the coordinate plane. Comparing two functions visually helps understand their behavior.

• Example: $f(x) = 2x$ and $g(x) = 2x - 3$ are both linear functions.

The blue line represents $f(x) = 2x$ and the red line represents $g(x) = 2x - 3$. Both are straight lines, but $g(x)$ is shifted downward by 3 units.

Analyzing Functions Using Graphs

Interpreting Values

To find $f(x)$ for a specific x, locate the x-value on the graph and read the corresponding y-value. To find x for a specific $f(x)$, locate the y-value and find the corresponding x-value(s).

This graph shows how the T cell count changes over time after HIV infection. The function $y = f(x)$ models the relationship between time and T cell count.

Identifying Domain and Range from a Graph

Graphical Approach

To find the domain, look for all x-values where the graph exists. To find the range, look for all y-values the graph attains.

For this graph, the domain is $[-2, 1]$ and the range is $[0, 3]$.

Example

• Domain: $(-2, 1]$ (open at -2, closed at 1)

• Range: $[-1, 2)$ (closed at -1, open at 2)

• f(1): Find the y-value when x = 1.

• f(-2): Find the y-value when x = -2.

Intercepts of Functions

x- and y-Intercepts

The x-intercepts (or zeros) are points where the graph crosses the x-axis ($y = 0$). The y-intercept is the point where the graph crosses the y-axis ($x = 0$). A function can have multiple x-intercepts but only one y-intercept.

• x-intercepts: Points where the graph crosses the x-axis.

• y-intercept: Point where the graph crosses the y-axis.

• Zeros: Values of x where $f(x) = 0$.

Example Table: Intercepts and Domain/Range

Feature	Description
x-intercepts	Where the graph crosses the x-axis
y-intercept	Where the graph crosses the y-axis
Domain	All x-values for which the function is defined
Range	All y-values the function attains

Summary Table: Function Properties

Property	How to Identify
Function	Each input has one output; passes vertical line test
Domain	All x-values with points on the graph
Range	All y-values with points on the graph
x-intercepts	Where graph crosses x-axis
y-intercept	Where graph crosses y-axis

Practice Questions

• Is the graph a function?

• Find the domain and range using interval notation.

• For how many values of x is $f(x) = 32$?

• Find one value of x such that $f(x) = 32$.

• Find $f(-2)$.

• What are the x-intercepts?

• Find the y-intercept.

• What are the zeros?

Additional info: These notes cover fundamental concepts from Precalculus Chapter 2: Functions and Graphs, including definitions, graphical analysis, and practice with domain, range, and intercepts.