Find the Domain and Range f(x) = square root of 4-2x

$f (x) = \sqrt{4 - 2 x}$

Step 1

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

$4 - 2 x \geq 0$

Step 2

Solve for $x$ .

Subtract $4$ from both sides of the inequality.

$- 2 x \geq - 4$

Divide each term in $- 2 x \geq - 4$ by $- 2$ and simplify.

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

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

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 4$ by $- 2$ .

$x \leq 2$

Step 3

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

Interval Notation:

$(- \infty, 2]$

Set-Builder Notation:

${x | x \leq 2}$

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: $(- \infty, 2], {x | x \leq 2}$

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

Step 6

Find the Domain and Range f(x) = square root of 4-2x

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