Find the Domain and Range f(x)=4 square root of 2x-1+3

$f (x) = 4 \sqrt{2 x - 1} + 3$ ?

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

$2 x - 1 \geq 0$

Solve for $x$ .

Add $1$ to both sides of the inequality.

$2 x \geq 1$

Divide each term in $2 x \geq 1$ by $2$ and simplify.

Divide each term in $2 x \geq 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Interval Notation:

$[\frac{1}{2}, \infty)$

Set-Builder Notation:

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

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

Interval Notation:

$[3, \infty)$

Set-Builder Notation:

${y | y \geq 3}$

Determine the domain and range.

Domain: $[\frac{1}{2}, \infty), {x | x \geq \frac{1}{2}}$

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