Graphs of Functions: Domain, Range, and Intercepts

Section 1.2 – The Graph of a Function

Learning Objectives

• Determine when a graph is a function.

• Identify information from or about a graph.

• Find the zeros or intercepts from or about a function and its graph.

Graphs of Functions

Definition of a Function

A function is a relation in which every input value (usually x) has exactly one output value (usually y). The graph of a function visually represents this relationship.

Vertical Line Test

The Vertical Line Test is a graphical method used to determine if a relation is 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.

• Passes the test: The graph is a function.

• Fails the test: The graph is not a function.

Example: Identifying Functions from Graphs

• Graph a) $y = x^4 + 1$: Passes the vertical line test; it is a function.

• Graph b) $y = \sqrt{x}$: Passes the vertical line test; it is a function.

• Graph c) $x^2 - y^2 = 1$: Fails the vertical line test; not a function.

• Graph d) $y = |x|$: Passes the vertical line test; it is a function.

Obtaining Information from a Graph

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined. The range is the set of all possible output values (y-values) the function can produce.

Set-Builder Notation

Set-builder notation describes a set based on a specific condition or rule. For example, the domain of $f(x) = \sqrt{x}$ can be written as $\{x \mid x \geq 0\}$.

Interval Notation

Interval notation expresses a continuous range of values. For example, $[0, \infty)$ means all real numbers from 0 to infinity, including 0 but not infinity.

• Open interval: $(a, b)$ excludes endpoints.

• Closed interval: $[a, b]$ includes endpoints.

• Half-open interval: $[a, b)$ or $(a, b]$ includes one endpoint.

Example: Determining Domain and Range from Graphs

• Graph a): Domain and range are determined by observing the leftmost, rightmost, lowest, and highest points on the graph.

• Graph b): Use set-builder and interval notation to express domain and range.

Intercepts of a Function

Y-Intercept

The y-intercept occurs when $x = 0$. To find the y-intercept, evaluate $f(0)$. The y-intercept is the point $(0, f(0))$.

X-Intercept(s)

The x-intercept(s) occur when the function equals zero, i.e., $f(x) = 0$. Solve for $x$ to find the x-intercepts, which are points $(x, 0)$.

Example: Finding Intercepts from a Graph

• Y-intercept: Find where the graph crosses the y-axis.

• X-intercept(s): Find where the graph crosses the x-axis.

Analyzing Graphs: Practice Problems

Example 3: Using a Graph to Identify Key Features

• Domain (interval notation): List all x-values for which the graph exists.

• Range (interval notation): List all y-values the graph attains.

• X-intercept(s): Identify all points where the graph crosses the x-axis.

• Y-intercept: Identify the point where the graph crosses the y-axis.

• Find $f(-2)$ and $f(5)$: Read the corresponding y-values from the graph at $x = -2$ and $x = 5$.

• How many times does the line $y = 2$ intersect the graph? Count the intersection points.

• For what values of $f(x) > 0$ (interval notation): Identify intervals where the graph is above the x-axis.

• For what values of $f(x) < 0$ (interval notation): Identify intervals where the graph is below the x-axis.

• Is $f(-2)$ negative or positive? Check the sign of the y-value at $x = -2$.

Try 1: Practice with a New Graph

• Domain (interval notation): Identify the set of x-values for which the graph is defined.

• Range (interval notation): Identify the set of y-values the graph attains.

• X-intercept(s): Find all points where the graph crosses the x-axis.

• Y-intercept: Find the point where the graph crosses the y-axis.

• Find $f(0)$ and $f(5)$: Read the y-values from the graph at $x = 0$ and $x = 5$.

Try 2: Analyzing a Linear Function

• Given $f(x) = -2x + 7$:

• Find the x-intercept(s): Set $f(x) = 0$ and solve for $x$: $0 = -2x + 7$ $2x = 7$ $x = \frac{7}{2}$

• Find the y-intercept(s): Evaluate $f(0)$: $f(0) = -2(0) + 7 = 7$ So the y-intercept is $(0, 7)$.

Summary Table: Notation for Domain and Range

Notation	Format	Example	Includes Endpoints?
Set-Builder	$\{x \mid x > 0\}$	All $x$ greater than 0	No
Interval (open)	$(a, b)$	$(0, 5)$	No
Interval (closed)	$[a, b]$	$[0, 5]$	Yes
Interval (half-open)	$[a, b)$ or $(a, b]$	$[0, 5)$	One endpoint

Additional info: These notes cover foundational concepts in College Algebra, including the definition of a function, graphical analysis, and notation for domain and range. Practice problems reinforce the application of these concepts to real graphs and equations.