Finding Domain and Range from Graphs

We will now return to our set of toolkit functions to determine the domain and range of each.

Figure 13 For the constant function $f(x)=c$, the domain consists of all real numbers; there are no restrictions on the input. The only output value is the constant $c$, so the range is the set $\{c\}$ that contains this single element. In interval notation, this is written as $[c, c]$, the interval that both begins and ends with $c$.

Figure 14 For the identity function $f(x)=x$, there is no restriction on $x$. Both the domain and range are the set of all real numbers.

Figure 15 For the absolute value function $f(x)=|x|$, there is no restriction on $x$. However, because absolute value is defined as a distance from $0$, the output can only be greater than or equal to $0$.

Figure 16 For the quadratic function $f(x)=x^{2}$, the domain is all real numbers since the horizontal extent of the graph is the whole real number line. Because the graph does not include any negative values for the range, the range is only nonnegative real numbers.

Figure 17 For the cubic function $f(x)=x^{3}$, the domain is all real numbers because the horizontal extent of the graph is the whole real number line. The same applies to the vertical extent of the graph, so the domain and range include all real numbers.

Figure 18 For the reciprocal function $f(x)=\frac{1}{x}$, we cannot divide by $0$, so we must exclude $0$ from the domain. Further, 1 divided by any value can never be $0$, so the range also will not include 0. In set-builder notation, we could also write $\{x \mid x \neq 0\}$, the set of all real numbers that are not zero.

Figure 19 For the reciprocal squared function $f(x)=\frac{1}{x^{2}}$, we cannot divide by 0, so we must exclude $0$ from the domain. There is also no $x$ that can give an output of $0$, so $0$ is excluded from the range as well. Note that the output of this function is always positive due to the square in the denominator, so the range includes only positive numbers.

Figure 20 For the square root function $f(x)=\sqrt{x}$, we cannot take the square root of a negative real number, so the domain must be $0$ or greater. The range also excludes negative numbers because the square root of a positive number $x$ is defined to be positive, even though the square of the negative number $-\sqrt{x}$ also gives us $x$.

Figure 21 For the cube root function $f(x)=\sqrt[3]{x}$, the domain and range include all real numbers. Note that there is no problem taking a cube root, or any oddinteger root, of a negative number, and the resulting output is negative (it is an odd function).

HOW TO

Given the formula for a function, determine the domain and range.

1. Exclude from the domain any input values that result in division by zero.
2. Exclude from the domain any input values that have nonreal (or undefined) number outputs.
3. Use the valid input values to determine the range of the output values.
4. Look at the function graph and table values to confirm the actual function behavior.

EXAMPLE 8

Finding the Domain and Range Using Toolkit Functions

Find the domain and range of $f(x)=2 x^{3}-x$.

Solution

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is $(-\infty, \infty)$ and the range is also $(-\infty, \infty)$.

EXAMPLE 9

Finding the Domain and Range

Find the domain and range of $f(x)=\frac{2}{x+1}$.

Solution

We cannot evaluate the function at $-1$ because division by zero is undefined. The domain is $(-\infty,-1) \cup(-1, \infty)$. Because the function is never zero, we exclude $0$ from the range. The range is $(-\infty, 0) \cup(0, \infty)$.

EXAMPLE 10

Finding the Domain and Range

Find the domain and range of $f(x)=2 \sqrt{x+4}$.

Solution

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

$x+4 \geq 0 \text { when } x \geq-4$

The domain of $f(x)$ is $[-4, \infty)$.

We then find the range. We know that $f(-4)=0$, and the function value increases as $x$ increases without any upper limit. We conclude that the range of $f$ is $[0, \infty)$.

Analysis

Figure 22 represents the function $f$.

Figure 22

TRY IT #7

Find the domain and range of $f(x)=-\sqrt{2-x}$.