Lesson 3: Graphing Radical Functions

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 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 $(1, \infty)$.

Explanation

To find a radical function's domain, check the index. For an even index, the expression inside the radical must be non-negative. For an odd index, the domain includes all real numbers because odd roots can handle any value.

Property

Examples