Precalculus Examples

$p (x) = 2 \sqrt{- \frac{1}{2} x + 4}$

Step 1

Set the radicand in $\sqrt{- \frac{1}{2} x + 4}$ greater than or equal to $0$ to find where the expression is defined.

$- \frac{1}{2} x + 4 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Combine $x$ and $\frac{1}{2}$ .

$- \frac{x}{2} + 4 \geq 0$

Step 2.2

Subtract $4$ from both sides of the inequality.

$- \frac{x}{2} \geq - 4$

Step 2.3

Divide each term in $- \frac{x}{2} \geq - 4$ by $- 1$ and simplify.

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Step 2.3.1

Divide each term in $- \frac{x}{2} \geq - 4$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- \frac{x}{2}}{- 1} \leq \frac{- 4}{- 1}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{x}{2}}{1} \leq \frac{- 4}{- 1}$

Step 2.3.2.2

Divide $\frac{x}{2}$ by $1$ .

$\frac{x}{2} \leq \frac{- 4}{- 1}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 4$ by $- 1$ .

$\frac{x}{2} \leq 4$

Step 2.4

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 \leq 4 \cdot 2$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

Cancel the common factor of $2$ .

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Step 2.5.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 \leq 4 \cdot 2$

Step 2.5.1.1.2

Rewrite the expression.

$x \leq 4 \cdot 2$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Multiply $4$ by $2$ .

$x \leq 8$

Step 3

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

Interval Notation:

$(- \infty, 8]$

Set-Builder Notation:

${x | x \leq 8}$

Step 4