Lesson 8.7: Use Radicals in Functions

New Concept

A radical function, like $f(x) = \sqrt{x}$, uses a radical to define its output. We'll learn how to evaluate these functions for specific inputs, determine the set of all possible inputs (the domain), and graph them.

What’s next

First, we'll dive into evaluating radical functions. Then, you'll work through interactive examples on finding their domains and graphing them step-by-step.

Property

A radical function is a function that is defined by a radical expression. To evaluate a radical function, we find the value of $f(x)$ for a given value of $x$ just as we did in our previous work with functions.

Examples

• For the function $f(x) = \sqrt{2x - 1}$, to find $f(5)$, substitute 5 for $x$: $f(5) = \sqrt{2(5) - 1} = \sqrt{9} = 3$.
• For the function $g(x) = \sqrt[3]{x - 6}$, to find $g(-2)$, substitute -2 for $x$: $g(-2) = \sqrt[3]{-2 - 6} = \sqrt[3]{-8} = -2$.
• For the function $f(x) = \sqrt[4]{5x - 4}$, evaluating $f(-12)$ gives $\sqrt[4]{-64}$, which is not a real number, so the function has no value at $x = -12$.

Explanation

A radical function is simply a function that contains a root, like a square root or cube root. To evaluate it, you just substitute the given number for the variable $x$ and then simplify the expression under the radical.

Property

When $n$ is an even number and:

• $a \geq 0$, then $\sqrt[n]{a}$ is a real number.
• $a < 0$, then $\sqrt[n]{a}$ is not a real number.

When $n$ is an odd number, $\sqrt[n]{a}$ is a real number for all values of $a$.

Examples

• For $\sqrt{36}$, the index $n=2$ is even and the radicand $a=36$ is positive, so the result, 6, is a real number.
• For $\sqrt{-16}$, the index $n=2$ is even but the radicand $a=-16$ is negative, so the result is not a real number.
• For $\sqrt[3]{-64}$, the index $n=3$ is odd, so it can have a negative radicand. The result, -4, is a real number.

Property

When the index of the radical is even, the radicand must be greater than or equal to zero.

When the index of the radical is odd, the radicand can be any real number.

Examples

• To find the domain of $f(x) = \sqrt{3x - 4}$, set the radicand $3x - 4 \geq 0$. Solving gives $x \geq \frac{4}{3}$. The domain is $[\frac{4}{3}, \infty)$.
• To find the domain of $f(x) = \sqrt[3]{2x^2 + 3}$, the index is odd, so the radicand can be any real number. The domain is $(-\infty, \infty)$.
• For $g(x) = \sqrt{\frac{6}{x-1}}$, the radicand must be positive. Since the numerator is positive, the denominator must be positive, so $x-1 > 0$. The domain is $(1, \infty)$.

Property

Before we graph any radical function, we first find the domain of the function. We choose $x$-values, substitute them in, and then create a chart of points to plot. After graphing, we can determine the range by observing the set of all possible $y$-values.

Examples

• For $f(x) = \sqrt{x + 3}$, the domain is $x \geq -3$, or $[-3, \infty)$. Key points are $(-3,0)$, $(-2,1)$, and $(1,2)$. The graph starts at $(-3,0)$ and moves up and to the right. The range is $[0, \infty)$.
• For $f(x) = \sqrt[3]{x}$, the domain is all real numbers, $(-\infty, \infty)$. Key points include $(-8,-2)$, $(-1,-1)$, $(0,0)$, $(1,1)$, and $(8,2)$. The graph passes through the origin and extends infinitely in both directions. The range is $(-\infty, \infty)$.
• For $f(x) = \sqrt{3-x}$, the domain requires $3-x \geq 0$, so $x \leq 3$. The domain is $(-\infty, 3]$. Key points are $(3,0)$, $(2,1)$, and $(-1,2)$. The range is $[0, \infty)$.

Explanation

To graph a radical function, first determine its domain. Then, select convenient x-values from the domain that make the radicand a perfect square or cube. Plot these points to sketch the graph and identify the range.