Find the Domain and Range f(x) = square root of |x|

$f (x) = \sqrt{| x |}$

Set the radicand in $\sqrt{| x |}$ greater than or equal to $0$ to find where the expression is defined.

$| x | \geq 0$

Solve for $x$ .

Write $| x | \geq 0$ as a piecewise.

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x \geq 0$

Write as a piecewise.

${\begin{cases} x \geq 0 & x \geq 0 \\ - x \geq 0 & x < 0 \end{cases}$

Find the intersection of $x \geq 0$ and $x \geq 0$ .

$x \geq 0$

Solve $- x \geq 0$ when $x < 0$ .

Divide each term in $- x \geq 0$ by $- 1$ and simplify.

Divide each term in $- x \geq 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{0}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{0}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{0}{- 1}$

Simplify the right side.

Divide $0$ by $- 1$ .

$x \leq 0$

Find the intersection of $x \leq 0$ and $x < 0$ .

$x < 0$

Find the union of the solutions.

All real numbers

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $[0, \infty), {y | y \geq 0}$

Find the Domain and Range f(x) = square root of |x|

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