## How to Find Domain and Range of Radical Functions

### Domain and Range of Functions

The domain of a function $$f(x)$$ is all the possible values of $$x$$ that can exist and still define the expression $$f(x).$$
The range of a function $$f(x)$$ is all the possible values that the expression $$f(x)$$ can have when $$x$$ is any number in the function's domain.

### Domain and Range of Radical Functions with Square Root

The domain and range of radical functions with square root, $$f(x) \ = \ \sqrt{x}$$ is $$[0 \ , \ ∞)$$.

In general, you can find the domain of a composite square root function $$\sqrt{g(x)}$$ by finding the values of $$x$$ that make $$g(x) \ ≥ \ 0$$. (The expression under the radicals can't be negative.)

### Example

Find the range and domain of $$f(x) \ = \ \sqrt{1 \ - \ x}$$

Solution:

Step1: We remember that having a negative number under the square root radical is not possible. So, to find the function's domain, make sure that the expression inside the square root is greater than or equal to zero:
$$1 \ - \ x \ ≥ \ 0 \ ⇒ \ -x \ ≥ \ -1 \ ⇒ \ x \ ≤ \ 1$$ or in interval notation: $$(-∞ \ , \ 1]$$
The range of $$f(x)$$ is $$[0 \ , \ ∞)$$.

Step2: A function's range is the set of all its possible values. We know that the square root function $$f(x) \ = \ \sqrt{x}$$ has a range of $$[0 \ , \ ∞)$$. In other words, for any number $$y$$ in the range $$[0 \ , \ ∞)$$, we can find a number $$x$$ that meets the condition satisfies.
Since, a function value of $$\sqrt{1 \ - \ x}$$ could be any number in the range $$[0 \ , \ ∞)$$, the range of $$f(x)$$ is $$[0 \ , \ ∞)$$.

### Domain and Range of Radical Functions with Cube Root

All of the numbers in the domain and range of the cube root function, $$f(x) \ = \ \sqrt[3]{x}$$ , are real numbers. This is represented by $$(-∞ \ , \ ∞)$$ or $$ℝ$$.

### Example

Find the domain of $$f(x) \ = \ \sqrt[3]{9 \ + \ 5x}$$

Solution

Remember that the cube root function's domain and range are $$(-∞ \ , \ ∞)$$. In other words, there are no domain restrictions on the cube root function. Since there are no restrictions on the domain of the expression $$9 \ + \ 5x$$, there are no limits on the possible $$x$$-values for this function. So, all real numbers are in the domain of this function.

### Exercises for Domain and Range of Radical Functions

1) Identify the domain and range: $$\sqrt{x \ - \ 5}$$

2) Identify the domain and range: $$\sqrt{x \ + \ 1} \ + \ 7$$

3) Identify the domain and range: $$\sqrt{x^2 \ - \ 4} \ - \ 3$$

4) Identify the domain and range: $$\sqrt{x^2 \ - \ 8} \ + \ 9$$

5) Identify the domain and range: $$\sqrt{x^2 \ + \ 9} \ - \ 13$$

6) Identify the domain and range: $$\sqrt{x^2 \ + \ 3} \ - \ 27$$

7) Identify the domain and range: $$\sqrt{x^2 \ + \ 18} \ + \ 4$$

8) Identify the domain and range: $$\sqrt{x^2 \ - \ 12} \ + \ 11$$

9) Identify the domain and range: $$\sqrt[3]{x^3 \ - \ 27} \ - \ 81$$

10) Identify the domain and range: $$\sqrt[3]{x \ + \ 9} \ - \ 54$$

1) Identify the domain and range: $$\sqrt{x \ - \ 5}$$

$$\color{red}{D: \ x \ - \ 5 \ ≥ \ 0 \ ⇒ \ x \ ≥ \ 5}$$
$$\color{red}{R: \ 0 \ ≤ \ y \ < \ ∞}$$

2) Identify the domain and range: $$\sqrt{x \ + \ 1} \ + \ 7$$

$$\color{red}{D: \ x \ + \ 1 \ ≥ \ 0 \ ⇒ \ x \ ≥ \ -1}$$
$$\color{red}{R: \ 7 \ ≤ \ y \ < \ ∞}$$

3) Identify the domain and range: $$\sqrt{x^2 \ - \ 4} \ - \ 3$$

$$\color{red}{D: \ x^2 \ - \ 4 \ ≥ \ 0 \ ⇒ \ -2 \ ≥ \ x \ ≥ \ 2}$$
$$\color{red}{R: \ -3 \ ≤ \ y \ < \ ∞}$$

4) Identify the domain and range: $$\sqrt{x^2 \ - \ 8} \ + \ 9$$

$$\color{red}{D: \ x^2 \ - \ 8 \ ≥ \ 0 \ ⇒ \ -2\sqrt{2} \ ≥ \ x \ ≥ \ 2\sqrt{2}}$$
$$\color{red}{R: \ 9 \ ≤ \ y \ < \ ∞}$$

5) Identify the domain and range: $$\sqrt{x^2 \ + \ 9} \ - \ 13$$

$$\color{red}{D: \ x^2 \ + \ 9 \ ≥ \ 0 \ ⇒ \ -∞ \ < \ x \ < \ ∞}$$
$$\color{red}{R: \ -13 \ ≤ \ y \ < \ ∞}$$

6) Identify the domain and range: $$\sqrt{x^2 \ + \ 3} \ - \ 27$$

$$\color{red}{D: \ x^2 \ + \ 3 \ ≥ \ 0 \ ⇒ \ -∞ \ < \ x \ < \ ∞}$$
$$\color{red}{R: \ -27 \ ≤ \ y \ < \ ∞}$$

7) Identify the domain and range: $$\sqrt{x^2 \ + \ 18} \ + \ 4$$

$$\color{red}{D: \ x^2 \ + \ 18 \ ≥ \ 0 \ ⇒ \ -∞ \ < \ x \ < \ ∞}$$
$$\color{red}{R: \ 4 \ ≤ \ y \ < \ ∞}$$

8) Identify the domain and range: $$\sqrt{x^2 \ - \ 12} \ + \ 11$$

$$\color{red}{D: \ x^2 \ - \ 12 \ ≥ \ 0 \ ⇒ \ -2\sqrt{3} \ ≥ \ x \ ≥ \ 2\sqrt{3}}$$
$$\color{red}{R: \ 11 \ ≤ \ y \ < \ ∞}$$

9) Identify the domain and range: $$\sqrt[3]{x^3 \ - \ 27} \ - \ 81$$

$$\color{red}{D: \ -∞ \ < \ x \ < \ ∞}$$
$$\color{red}{R: \ -∞ \ < \ y \ < \ ∞}$$

10) Identify the domain and range: $$\sqrt[3]{x \ + \ 9} \ - \ 54$$

$$\color{red}{D: \ -∞ \ < \ x \ < \ ∞}$$
$$\color{red}{R: \ -∞ \ < \ y \ < \ ∞}$$

## Domain and Range of Radical Functions Practice Quiz

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