The Range of a Function

The Basic Concept of a Range

For a function $y = f(x)$, the range is the set of all the possible values of $y$.

Methods for Finding the Range of a Function

1. The Direct Method (by Inspection)

For a simple function, you can read the range off the properties of the function.

Example 1: find the range of $f(x) = x^2$ ($x \in \mathbb{R}$).

Solution: $x^2 \geq 0$ holds for every real $x$, and as $x$ ranges over all real numbers $x^2$ takes every non-negative real value.

Hence the range is $[0, +\infty)$.

2. Completing the Square

For a quadratic function, completing the square finds the extremum of the function and so determines the range.

Example 2: find the range of $f(x) = x^2 - 4x + 5$ ($x \in \mathbb{R}$).

Solution: complete the square: $f(x) = x^2 - 4x + 5 = (x-2)^2 + 1$

Since $(x-2)^2 \geq 0$, we have $f(x) = (x-2)^2 + 1 \geq 1$.

When $x = 2$, $f(x)$ attains its minimum value of 1.

Hence the range is $[1, +\infty)$.

3. The Inverse Function Method

Put $y = f(x)$, solve for $x = g(y)$, and then use the domain of $x$ to determine the range of $y$.

Example 3: find the range of $f(x) = \frac{x+1}{x-2}$ ($x \neq 2$).

Solution: put $y = \frac{x+1}{x-2}$ and solve the equation for $x$:

$y(x-2) = x+1$ $yx - 2y = x + 1$ $yx - x = 2y + 1$ $x(y-1) = 2y + 1$

When $y \neq 1$, $x = \frac{2y+1}{y-1}$.

Since the domain of the original function requires $x \neq 2$, we have: $\frac{2y+1}{y-1} \neq 2$ $2y + 1 \neq 2(y-1)$ $2y + 1 \neq 2y - 2$ $1 \neq -2$

This inequality always holds, so $y$ can take every real number except 1.

Hence the range is $(-\infty, 1) \cup (1, +\infty)$.

4. The Monotonicity Method

Use the monotonicity of the function to determine its extremum on a given interval, and hence find the range.

Example 4: find the range of $f(x) = 2x - 1$ ($x \in [1, 3]$).

Solution: the function $f(x) = 2x - 1$ is increasing.

On the interval $[1, 3]$:

• when $x = 1$, $f(1) = 2 \times 1 - 1 = 1$
• when $x = 3$, $f(3) = 2 \times 3 - 1 = 5$

Hence the range is $[1, 5]$.

5. The Graphical Method

Determine the range of the function values by looking at the graph of the function.

Example 5: find the range of $f(x) = \sqrt{4-x^2}$.

Solution: first find the domain: $4 - x^2 \geq 0$, that is, $x^2 \leq 4$, so $x \in [-2, 2]$.

Since $0 \leq 4 - x^2 \leq 4$ (for $x \in [-2, 2]$), we have: $0 \leq \sqrt{4-x^2} \leq 2$

• when $x = \pm 2$, $f(x) = 0$
• when $x = 0$, $f(x) = 2$

Hence the range is $[0, 2]$.

The Ranges of Common Functions

Exercises

Exercise 1

The sales volume $Q$ (in units) of a commodity and its price $P$ (in yuan) satisfy the functional relationship $Q = 200 - 10P$, where the price $P$ ranges over $[5, 15]$ yuan.

1. Find the domain and the range of this function
2. When the price is 8 yuan, what is the sales volume?
3. To keep the sales volume at no less than 100 units, what range should the price be held to?

Exercise 2

Use completing the square to find the range of $f(x) = -x^2 + 4x - 1$.

Exercise 3

Use the inverse function method to find the range of $f(x) = \frac{2x - 1}{x + 3}$ ($x \neq -3$).

Exercise 4

Given the table of values of a function below, write down its analytical expression, and find its domain and range.

Summary

Symbols Used in This Article