Find the Domain and Range y = square root of 1/2x+5

$y = \sqrt{\frac{1}{2} x} + 5$

Step 1

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

$\frac{1}{2} x \geq 0$

Step 2

Multiply each term in $\frac{1}{2} x \geq 0$ by $2$ to eliminate the fractions.

Multiply each term in $\frac{1}{2} x \geq 0$ by $2$ .

$\frac{1}{2} x \cdot 2 \geq 0 \cdot 2$

Simplify the left side.

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

$\frac{x}{2} \cdot 2 \geq 0 \cdot 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x}{2} \cdot 2 \geq 0 \cdot 2$

Rewrite the expression.

$x \geq 0 \cdot 2$

Simplify the right side.

Multiply $0$ by $2$ .

$x \geq 0$

Step 3

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 4

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

Interval Notation:

$[5, \infty)$

Set-Builder Notation:

${y | y \geq 5}$

Step 5

Determine the domain and range.

Domain: $[0, \infty), {x | x \geq 0}$

Range: $[5, \infty), {y | y \geq 5}$

Step 6