Find the Inverse f(x) = square root of x+2+9

$f (x) = \sqrt{x + 2 + 9}$

Write $f (x) = \sqrt{x + 2 + 9}$ as an equation.

$y = \sqrt{x + 2 + 9}$

Interchange the variables.

$x = \sqrt{y + 2 + 9}$

Solve for $y$ .

Rewrite the equation as $\sqrt{y + 2 + 9} = x$ .

$\sqrt{y + 2 + 9} = x$

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

${\sqrt{y + 2 + 9}}^{2} = x^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y + 2 + 9}$ as ${(y + 2 + 9)}^{\frac{1}{2}}$ .

${({(y + 2 + 9)}^{\frac{1}{2}})}^{2} = x^{2}$

Simplify the left side.

Simplify ${({(y + 2 + 9)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(y + 2 + 9)}^{\frac{1}{2}})}^{2}$ .

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

${(y + 2 + 9)}^{\frac{1}{2} \cdot 2} = x^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(y + 2 + 9)}^{\frac{1}{2} \cdot 2} = x^{2}$

Rewrite the expression.

${(y + 2 + 9)}^{1} = x^{2}$

Add $2$ and $9$ .

${(y + 11)}^{1} = x^{2}$

Simplify.

$y + 11 = x^{2}$

Subtract $11$ from both sides of the equation.

$y = x^{2} - 11$

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

$f^{- 1} (x) = x^{2} - 11$

Verify if $f^{- 1} (x) = x^{2} - 11$ is the inverse of $f (x) = \sqrt{x + 2 + 9}$ .

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (\sqrt{x + 2 + 9})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{x + 2 + 9}) = {(\sqrt{x + 2 + 9})}^{2} - 11$

Simplify each term.

Add $2$ and $9$ .

$f^{- 1} (\sqrt{x + 2 + 9}) = {\sqrt{x + 11}}^{2} - 11$

Rewrite ${\sqrt{x + 11}}^{2}$ as $x + 11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 11}$ as ${(x + 11)}^{\frac{1}{2}}$ .

$f^{- 1} (\sqrt{x + 2 + 9}) = {({(x + 11)}^{\frac{1}{2}})}^{2} - 11$

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

$f^{- 1} (\sqrt{x + 2 + 9}) = {(x + 11)}^{\frac{1}{2} \cdot 2} - 11$

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

$f^{- 1} (\sqrt{x + 2 + 9}) = {(x + 11)}^{\frac{2}{2}} - 11$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\sqrt{x + 2 + 9}) = {(x + 11)}^{\frac{2}{2}} - 11$

Rewrite the expression.

$f^{- 1} (\sqrt{x + 2 + 9}) = (x + 11) - 11$

Simplify.

$f^{- 1} (\sqrt{x + 2 + 9}) = x + 11 - 11$

Combine the opposite terms in $x + 11 - 11$ .

Subtract $11$ from $11$ .

$f^{- 1} (\sqrt{x + 2 + 9}) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\sqrt{x + 2 + 9}) = x$

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (x^{2} - 11)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (x^{2} - 11) = \sqrt{(x^{2} - 11) + 2 + 9}$

Add $- 11$ and $2$ .

$f (x^{2} - 11) = \sqrt{x^{2} - 9 + 9}$

Add $- 9$ and $9$ .

$f (x^{2} - 11) = \sqrt{x^{2} + 0}$

Add $x^{2}$ and $0$ .

$f (x^{2} - 11) = \sqrt{x^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$f (x^{2} - 11) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = x^{2} - 11$ is the inverse of $f (x) = \sqrt{x + 2 + 9}$ .

$f^{- 1} (x) = x^{2} - 11$

Find the Inverse f(x) = square root of x+2+9

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