Find the Domain and Range f(x)=5/2 square root of -(x-6)

$f (x) = \frac{5}{2} \sqrt{- (x - 6)}$

Step 1

Set the radicand in $\sqrt{- (x - 6)}$ greater than or equal to $0$ to find where the expression is defined.

$- (x - 6) \geq 0$

Step 2

Solve for $x$ .

Simplify $- (x - 6)$ .

Apply the distributive property.

$- x - - 6 \geq 0$

Multiply $- 1$ by $- 6$ .

$- x + 6 \geq 0$

Subtract $6$ from both sides of the inequality.

$- x \geq - 6$

Divide each term in $- x \geq - 6$ by $- 1$ and simplify.

Divide each term in $- x \geq - 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} \leq \frac{- 6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} \leq \frac{- 6}{- 1}$

Divide $x$ by $1$ .

$x \leq \frac{- 6}{- 1}$

Simplify the right side.

Divide $- 6$ by $- 1$ .

$x \leq 6$

Step 3

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

Interval Notation:

$(- \infty, 6]$

Set-Builder Notation:

${x | x \leq 6}$

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: $(- \infty, 6], {x | x \leq 6}$

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

Step 6

Find the Domain and Range f(x)=5/2 square root of -(x-6)

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