4.12.3 Range: The Output of a Function

Extending Your Thinking

Naming Input-Output Pairs

A graph is yet another way that a relation can be represented. The set of ordered pairs of all the points plotted is the relation. The set of all $x$-coordinates, or the input values, is the domain of the relation, and the set of all $y$-coordinates, or outputs, is the range.

Recall a coordinate is written $(x,y)$.

Name the input-output pairs for the graph below:

The input values are the $x$-values, and the output values are $y$-values. (1,3) (2,6) (3,12) (4,24)

This means the domain for the graphed relation is {1, 2, 3, 4} and the range is {3, 6, 12, 24}

Try it

Try It: Naming Input-Output Pairs

Name the input-output pairs of the graph below:

Here is how to write the input-output pairs:

$(1,4.5)$

$(2,9)$

$(3,18)$

$(4,36)$

Using Inequalities to Determine Range

The domain and range of a function describe the set of possible input values (domain) and output values (range) for that function. The range represents the values that the function can output. Inequalities can be used to represent the range of a function. Remember to consider the nature of the function, its properties, and any specific restrictions when determining the range using inequalities. Graphing the function can often provide visual clues to help confirm your analysis.

Use the following information to determine the range of a function using inequalities:

• Analyze the behavior and possible output values of the function, usually denoted as $f(x)$.
• Consider the range of the function based on its graph or algebraic properties.
• Identify the minimum and maximum possible values that the function can reach.

Use the following information to express the range of a function using inequalities:

• Use interval notation or set notation to represent the valid output values for the function.
• Use inequalities to express the minimum and maximum possible values. For example, if $x$ cannot be negative, you would write $(x\ge 0)$ or $[0,+\infty )$ in interval notation. For example, if the function's graph never goes below −3 and can reach any positive value, you would write $(-3<y<+\infty )$ as an inequality.

Example

Determine the range of this function: $p(x)=3,x\ge -2.$

Since the function $p(x)$ is constant and always equal to 3, the range of the function is a single value, which is 3.

Therefore, the range of $p(x)=3$, $x\ge -2$ is the set {3}.