Find the Domain and Range m(x)=(x+1)^(1/2)+2

$m (x) = {(x + 1)}^{\frac{1}{2}} + 2$

Step 1

Convert expressions with fractional exponents to radicals.

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$\sqrt{{(x + 1)}^{1}} + 2$

Anything raised to $1$ is the base itself.

$\sqrt{x + 1} + 2$

Step 2

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

$x + 1 \geq 0$

Step 3

Subtract $1$ from both sides of the inequality.

$x \geq - 1$

Step 4

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

Interval Notation:

$[- 1, \infty)$

Set-Builder Notation:

${x | x \geq - 1}$

Step 5

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

Interval Notation:

$[2, \infty)$

Set-Builder Notation:

${y | y \geq 2}$

Step 6

Determine the domain and range.

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

Range: $[2, \infty), {y | y \geq 2}$

Step 7