Find the Domain and Range f(x)=3+ square root of 2-4x

$f (x) = 3 + \sqrt{2 - 4 x}$

Step 1

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

$2 - 4 x \geq 0$

Step 2

Solve for $x$ .

Subtract $2$ from both sides of the inequality.

$- 4 x \geq - 2$

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

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

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

Simplify the left side.

Cancel the common factor of $- 4$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $- 2$ and $- 4$ .

Factor $- 2$ out of $- 2$ .

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

Cancel the common factors.

Factor $- 2$ out of $- 4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

Interval Notation:

$(- \infty, \frac{1}{2}]$

Set-Builder Notation:

${x | x \leq \frac{1}{2}}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[3, \infty)$

Set-Builder Notation:

${y | y \geq 3}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \frac{1}{2}], {x | x \leq \frac{1}{2}}$

Range: $[3, \infty), {y | y \geq 3}$

Step 6

Find the Domain and Range f(x)=3+ square root of 2-4x

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