Find the Inverse f(x) = square root of x-2 ; for x>=2

$f (x) = \sqrt{x - 2}$ ; for $x \geq 2$

Find the range of the given function.

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

$[0, \infty)$

Convert $[0, \infty)$ to an inequality.

$y \geq 0$

Find the inverse.

Interchange the variables.

$x = \sqrt{y - 2}$

Solve for $y$ .

Rewrite the equation as $\sqrt{y - 2} = x$ .

$\sqrt{y - 2} = x$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{y - 2}}^{2} = x^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y - 2}$ as ${(y - 2)}^{\frac{1}{2}}$ .

${({(y - 2)}^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${({(y - 2)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(y - 2)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(y - 2)}^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(y - 2)}^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

${(y - 2)}^{1} = x^{2}$

Simplify.

$y - 2 = x^{2}$

Add $2$ to both sides of the equation.

$y = x^{2} + 2$

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = x^{2} + 2$

Find the inverse using the domain and the range of the original function.

$f^{- 1} (x) = x^{2} + 2, x \geq 0$