A4

Functions and Their Features

Definition of a Function

A function is a relation that assigns each input value (from the domain) exactly one output value (from the range). Functions can be represented by equations, tables, or graphs.

• Domain: The set of all possible input values (x-values) for the function.

• Range: The set of all possible output values (y-values) for the function.

• Independent Variable: The input variable, usually denoted as x.

• Dependent Variable: The output variable, usually denoted as y or f(x).

Example: For $f(x) = 2x - 3$, the domain is all real numbers, and the range is also all real numbers.

Features of a Graph

Key Vocabulary

• x-intercept: The point(s) where the graph crosses the x-axis. Found by setting $f(x) = 0$ and solving for x.

• y-intercept: The point where the graph crosses the y-axis. Found by evaluating $f(0)$.

• Intervals of Increase: Where the function values rise as x increases.

• Intervals of Decrease: Where the function values fall as x increases.

• Intervals Where Positive: Where $f(x) > 0$.

• Intervals Where Negative: Where $f(x) < 0$.

Example: For the graph of $f(x)$ shown, the domain might be $(-, 6)$ and the range $(-6, 6)$, depending on the graph's extent.

Identifying Features from Graphs and Tables

Using Graphs

To identify features from a graph:

• Read the domain and range from the x- and y-axes.

• Find intercepts by locating where the graph crosses the axes.

• Determine intervals of increase/decrease by observing the graph's slope.

• Identify where the function is positive or negative by checking above/below the x-axis.

Example: If the graph increases from $x = -6$ to $x = 0$, then the interval of increase is $(-6, 0)$.

Using Tables

Tables list input-output pairs for a function. Features can be identified by:

• Domain: All listed x-values.

• Range: All listed f(x) values.

• Intervals of increase/decrease: Compare consecutive f(x) values.

• Positive/negative intervals: Note where f(x) is above or below zero.

x	f(x)
-2	2
-1	1
0	0
1	-1
2	-2

Additional info: The table above shows a function decreasing as x increases.

Piecewise Functions

Definition and Graphing

A piecewise function is defined by different expressions over different intervals of the domain.

• Each "piece" applies to a specific interval.

• Graph each piece separately, then combine for the full graph.

Example:

$f(x) = \begin{cases} 2x + 3 & \text{if } x < -2 \\ -\frac{1}{2}x + 2 & \text{if } x \geq -2 \end{cases}$

To graph, plot each piece on its interval, noting endpoints and intercepts.

Worked Examples

Example 1: Graph Features

• Domain: $(-6, 6)$

• Range: $(-6, 6)$

• x-intercept: $(2, 0)$

• y-intercept: $(0, 2)$

• Interval of Increase: $(-6, 0)$

• Interval of Decrease: $(0, 6)$

• Interval Where Positive: $(-6, 2)$

• Interval Where Negative: $(2, 6)$

Example 2: Table Features

• Domain: $[-2, 2]$

• Range: $[0, 4]$

• Interval of Increase: $(-2, 0)$

• Interval of Decrease: $(0, 2)$

• Interval Where Positive: $(-2, 2)$

• Interval Where Negative: None

Example 3: Piecewise Function Graph

Given $f(x)$ as a piecewise function, create a table of values for each interval, plot the graph, and identify features:

• Domain: $(-\infty, \infty)$

• Range: $(-\infty, 7]$

• x-intercept: $(0, 0)$

• y-intercept: $(0, 2)$

• Interval Where Positive: $(-\infty, 0)$

• Interval Where Negative: $(0, \infty)$

• Interval of Increase: $(-\infty, -2)$

• Interval of Decrease: $(-2, \infty)$

Summary Table: Features of Functions

Feature	How to Find	Example
Domain	Set of all x-values	$(-\infty, \infty)$
Range	Set of all f(x) values	$(-\infty, 7]$
x-intercept	Set $f(x) = 0$	$(0, 0)$
y-intercept	Find $f(0)$	$(0, 2)$
Intervals of Increase	Where graph rises	$(-\infty, -2)$
Intervals of Decrease	Where graph falls	$(-2, \infty)$
Intervals Positive	Where $f(x) > 0$	$(-\infty, 0)$
Intervals Negative	Where $f(x) < 0$	$(0, \infty)$

Conclusion

Understanding the features of functions and their graphs is essential in College Algebra. By analyzing equations, tables, and graphs, students can determine domains, ranges, intercepts, and intervals of increase/decrease and positivity/negativity. These skills are foundational for further study in mathematics and its applications.