Algebra Examples

$f (x) = {(- x)}^{\frac{1}{2}}$

Step 1

Convert expressions with fractional exponents to radicals.

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Step 1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt{{(- x)}^{1}}$

Step 1.2

Anything raised to $1$ is the base itself.

$\sqrt{- x}$

Step 2

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

$- x \geq 0$

Step 3

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

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Step 3.1

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.

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2

Divide $x$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0$ by $- 1$ .

$x \leq 0$

Step 4

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0]$

Set-Builder Notation:

${x | x \leq 0}$

Step 5

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${y | y \geq 0}$

Step 6

Determine the domain and range.

Domain: $(- \infty, 0], {x | x \leq 0}$

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

Step 7