Find the Domain of the Composition g(f(x)) f(x)=2x+1 , g(x) = square root of x

$f (x) = 2 x + 1$ , $g (x) = \sqrt{x}$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (2 x + 1)$ by substituting in the value of $f$ into $g$ .

$g (2 x + 1) = \sqrt{2 x + 1}$

Step 3

Remove parentheses.

$\sqrt{2 x + 1}$

Step 4

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

$2 x + 1 \geq 0$

Step 5

Solve for $x$ .

Subtract $1$ from 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 $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Step 6

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

Interval Notation:

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

Set-Builder Notation:

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

Step 7