Range of a Function

Article Contents

What is the Range?
Range vs Domain
Finding the Range from a Graph
Range of Common Functions
Finding the Range from a Formula
Worked Examples
Common Mistakes
Interactive Exercises

1. What is the Range? {#what-is-range}

The range of a function $f$ (written $R (f)$ or $H (f)$ ) is the set of all possible outputs -- all values that $f (x)$ can take.

R (f) = {y \in R ∣ y = f (x) for some x \in D (f)}

Think of it this way

The domain answers "what can I put in?" The range answers "what can come out?"

2. Range vs Domain {#range-vs-domain}

	Domain $D (f)$	Range $R (f)$
What?	Set of allowed inputs	Set of possible outputs
Variable	$x$ (independent)	$y = f (x)$ (dependent)
On graph	Horizontal extent	Vertical extent
Question	Which $x$ -values?	Which $y$ -values?

3. Finding the Range from a Graph {#range-from-graph}

On a graph, the range is the set of all $y$ -values that the graph reaches.

Method: Project the graph onto the y-axis. The range is the vertical "shadow" of the graph.

4. Range of Common Functions {#range-common}

Function	Formula	Range
Linear	$f (x) = k x + q$ ( $k \neq = 0$ )	$R (f) = R$
Constant	$f (x) = c$	$R (f) = {c}$
Direct proportion	$f (x) = k x$ ( $k \neq = 0$ )	$R (f) = R$
Inverse proportion	$f (x) = \frac{k}{x}$ ( $k \neq = 0$ )	$R (f) = R ∖ {0}$
Quadratic (basic)	$f (x) = x^{2}$	$R (f) = [0, + \infty)$
Quadratic (negative)	$f (x) = - x^{2}$	$R (f) = (- \infty, 0]$
Square root	$f (x) = x$	$R (f) = [0, + \infty)$
Absolute value	$f(x) =	x	$	$R (f) = [0, + \infty)$

5. Finding the Range from a Formula {#range-from-formula}

Method for simple functions

Ask: "What values can $f (x)$ produce?"

Example 1: $f (x) = x^{2} + 1$

Since $x^{2} \geq 0$ for all $x$ , we have:

f (x) = x^{2} + 1 \geq 0 + 1 = 1

The minimum value is 1 (when $x = 0$ ), and $f (x)$ can grow without bound.

R (f) = [1, + \infty)

Example 2: $f (x) = - x^{2} + 4$

Since $x^{2} \geq 0$ , we have $- x^{2} \leq 0$ , so:

f (x) = - x^{2} + 4 \leq 4

The maximum value is 4 (when $x = 0$ ), and $f (x)$ can decrease without bound.

R (f) = (- \infty, 4]

Example 3: $f (x) = \frac{2}{x}$

Since $x \neq = 0$ , the fraction $\frac{2}{x}$ can be any number except 0.

R (f) = R ∖ {0}

6. Worked Examples {#worked-examples}

Example A

Find the range of $f (x) = 3 x - 6$ on the domain $D (f) = R$ .

This is a linear function with slope $k = 3 \neq = 0$ . A non-constant linear function takes all real values.

R (f) = R

Example B

Find the range of $f (x) = ∣ x - 2∣$ .

The absolute value is always $\geq 0$ . When $x = 2$ , $f (2) = ∣2 - 2∣ = 0$ .

R (f) = [0, + \infty)

Example C

Find the range of $f (x) = x - 1 + 3$ .

Since $x - 1 \geq 0$ , we have $f (x) \geq 0 + 3 = 3$ .

When $x = 1$ : $f (1) = 0 + 3 = 3$ (minimum).

R (f) = [3, + \infty)

7. Common Mistakes {#common-mistakes}

Mistake	Correction
Confusing domain and range	Domain = inputs ( $x$ -values), Range = outputs ( $y$ -values)
Saying $R (f) = R$ for $f (x) = x^{2}$	$x^{2}$ is always $\geq 0$ , so $R (f) = [0, + \infty)$
Forgetting that $\frac{k}{x}$ never equals 0	$\frac{k}{x} = 0$ has no solution, so $0 \in / R (f)$
Not checking the minimum/maximum	For quadratics, find the vertex value first

8. Interactive Exercises {#interactive-exercises}

Practice finding ranges:

Range from a Graph

Summary

Concept	Description
Range $R (f)$	Set of all possible output values
From a graph	Project onto the y-axis -- the vertical extent
Linear ( $k \neq = 0$ )	$R (f) = R$
$f (x) = x^{2}$	$R (f) = [0, + \infty)$
$f (x) = k / x$	$R (f) = R ∖ {0}$

Domain of a Function -- which inputs are allowed
Functions and Graphs -- visualising functions
Introduction to Functions -- function basics
Linear Function -- range of linear functions