Algebra Examples

$f (x) = \sqrt{8 - x}$

Step 1

Set the radicand in

$\sqrt{8 - x}$ greater than or equal to

$0$ to find where the expression is defined.

$8 - x \geq 0$

Step 2

Solve for

$x$ .

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Step 2.1

Subtract

$8$ from both sides of the inequality.

$- x \geq - 8$

Step 2.2

Divide each term in

$- x \geq - 8$ by

$- 1$ and simplify.

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Step 2.2.1

Divide each term in

$- x \geq - 8$ by

$- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 8}{- 1}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 8}{- 1}$

Step 2.2.2.2

Divide

$x$ by

$1$ .

$x \leq \frac{- 8}{- 1}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide

$- 8$ by

$- 1$ .

$x \leq 8$

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

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 5

Determine the domain and range.

Domain:

$(- \infty, 8], {x | x \leq 8}$

Range:

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

Step 6