Find the Domain y=(5-x)^(1/2)

$y = {(5 - x)}^{\frac{1}{2}}$

Step 1

Convert expressions with fractional exponents to radicals.

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

$\sqrt{{(5 - x)}^{1}}$

Anything raised to $1$ is the base itself.

$\sqrt{5 - x}$

Step 2

Set the radicand in $\sqrt{5 - x}$ greater than or equal to $0$ to find where the expression is defined.

$5 - x \geq 0$

Step 3

Solve for $x$ .

Subtract $5$ from both sides of the inequality.

$- x \geq - 5$

Divide each term in $- x \geq - 5$ by $- 1$ and simplify.

Divide each term in $- x \geq - 5$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 5}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 5}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{- 5}{- 1}$

Simplify the right side.

Divide $- 5$ by $- 1$ .

$x \leq 5$

Step 4

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

Interval Notation:

$(- \infty, 5]$

Set-Builder Notation:

${x | x \leq 5}$

Step 5

Find the Domain y=(5-x)^(1/2)

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