Find the Inverse square root of 2-x+6

$\sqrt{2 - x} + 6$

Step 1

Interchange the variables.

$x = \sqrt{2 - y} + 6$

Step 2

Solve for $y$ .

Rewrite the equation as $\sqrt{2 - y} + 6 = x$ .

$\sqrt{2 - y} + 6 = x$

Subtract $6$ from both sides of the equation.

$\sqrt{2 - y} = x - 6$

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

${\sqrt{2 - y}}^{2} = {(x - 6)}^{2}$

Simplify each side of the equation.

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

${({(2 - y)}^{\frac{1}{2}})}^{2} = {(x - 6)}^{2}$

Simplify the left side.

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

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

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

${(2 - y)}^{\frac{1}{2} \cdot 2} = {(x - 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 - y)}^{\frac{1}{2} \cdot 2} = {(x - 6)}^{2}$

Rewrite the expression.

${(2 - y)}^{1} = {(x - 6)}^{2}$

Simplify.

$2 - y = {(x - 6)}^{2}$

Simplify the right side.

Simplify ${(x - 6)}^{2}$ .

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$2 - y = (x - 6) (x - 6)$

Expand $(x - 6) (x - 6)$ using the FOIL Method.

Apply the distributive property.

$2 - y = x (x - 6) - 6 (x - 6)$

Apply the distributive property.

$2 - y = x \cdot x + x \cdot - 6 - 6 (x - 6)$

Apply the distributive property.

$2 - y = x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$2 - y = x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6$

Move $- 6$ to the left of $x$ .

$2 - y = x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6$

Multiply $- 6$ by $- 6$ .

$2 - y = x^{2} - 6 x - 6 x + 36$

Subtract $6 x$ from $- 6 x$ .

$2 - y = x^{2} - 12 x + 36$

Solve for $y$ .

Move all terms not containing $y$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

$- y = x^{2} - 12 x + 36 - 2$

Subtract $2$ from $36$ .

$- y = x^{2} - 12 x + 34$

Divide each term in $- y = x^{2} - 12 x + 34$ by $- 1$ and simplify.

Divide each term in $- y = x^{2} - 12 x + 34$ by $- 1$ .

$\frac{- y}{- 1} = \frac{x^{2}}{- 1} + \frac{- 12 x}{- 1} + \frac{34}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{x^{2}}{- 1} + \frac{- 12 x}{- 1} + \frac{34}{- 1}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{- 1} + \frac{- 12 x}{- 1} + \frac{34}{- 1}$

Simplify the right side.

Simplify each term.

Move the negative one from the denominator of $\frac{x^{2}}{- 1}$ .

$y = - 1 \cdot x^{2} + \frac{- 12 x}{- 1} + \frac{34}{- 1}$

Rewrite $- 1 \cdot x^{2}$ as $- x^{2}$ .

$y = - x^{2} + \frac{- 12 x}{- 1} + \frac{34}{- 1}$

Move the negative one from the denominator of $\frac{- 12 x}{- 1}$ .

$y = - x^{2} - 1 \cdot (- 12 x) + \frac{34}{- 1}$

Rewrite $- 1 \cdot (- 12 x)$ as $- (- 12 x)$ .

$y = - x^{2} - (- 12 x) + \frac{34}{- 1}$

Multiply $- 12$ by $- 1$ .

$y = - x^{2} + 12 x + \frac{34}{- 1}$

Divide $34$ by $- 1$ .

$y = - x^{2} + 12 x - 34$

Step 3

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

$f^{- 1} (x) = - x^{2} + 12 x - 34$

Step 4

Verify if $f^{- 1} (x) = - x^{2} + 12 x - 34$ is the inverse of $f (x) = \sqrt{2 - x} + 6$ .

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 $f^{- 1} (\sqrt{2 - x} + 6)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - {(\sqrt{2 - x} + 6)}^{2} + 12 (\sqrt{2 - x} + 6) - 34$

Simplify each term.

Rewrite ${(\sqrt{2 - x} + 6)}^{2}$ as $(\sqrt{2 - x} + 6) (\sqrt{2 - x} + 6)$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - ((\sqrt{2 - x} + 6) (\sqrt{2 - x} + 6)) + 12 (\sqrt{2 - x} + 6) - 34$

Expand $(\sqrt{2 - x} + 6) (\sqrt{2 - x} + 6)$ using the FOIL Method.

Apply the distributive property.

$f^{- 1} (\sqrt{2 - x} + 6) = - (\sqrt{2 - x} (\sqrt{2 - x} + 6) + 6 (\sqrt{2 - x} + 6)) + 12 (\sqrt{2 - x} + 6) - 34$

Apply the distributive property.

$f^{- 1} (\sqrt{2 - x} + 6) = - (\sqrt{2 - x} \sqrt{2 - x} + \sqrt{2 - x} \cdot 6 + 6 (\sqrt{2 - x} + 6)) + 12 (\sqrt{2 - x} + 6) - 34$

Apply the distributive property.

$f^{- 1} (\sqrt{2 - x} + 6) = - (\sqrt{2 - x} \sqrt{2 - x} + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{2 - x} \sqrt{2 - x}$ .

Raise $\sqrt{2 - x}$ to the power of $1$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - (\sqrt{2 - x} \sqrt{2 - x} + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

Raise $\sqrt{2 - x}$ to the power of $1$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - (\sqrt{2 - x} \sqrt{2 - x} + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f^{- 1} (\sqrt{2 - x} + 6) = - ({\sqrt{2 - x}}^{1 + 1} + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

Add $1$ and $1$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - ({\sqrt{2 - x}}^{2} + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

Rewrite ${\sqrt{2 - x}}^{2}$ as $2 - x$ .

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

$f^{- 1} (\sqrt{2 - x} + 6) = - ({({(2 - x)}^{\frac{1}{2}})}^{2} + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

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

$f^{- 1} (\sqrt{2 - x} + 6) = - ({(2 - x)}^{\frac{1}{2} \cdot 2} + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

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

$f^{- 1} (\sqrt{2 - x} + 6) = - ({(2 - x)}^{\frac{2}{2}} + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\sqrt{2 - x} + 6) = - ({(2 - x)}^{\frac{2}{2}} + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

Rewrite the expression.

$f^{- 1} (\sqrt{2 - x} + 6) = - ((2 - x) + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

Simplify.

$f^{- 1} (\sqrt{2 - x} + 6) = - (2 - x + \sqrt{2 - x} \cdot 6 + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

Move $6$ to the left of $\sqrt{2 - x}$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - (2 - x + 6 \cdot \sqrt{2 - x} + 6 \sqrt{2 - x} + 6 \cdot 6) + 12 (\sqrt{2 - x} + 6) - 34$

Multiply $6$ by $6$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - (2 - x + 6 \sqrt{2 - x} + 6 \sqrt{2 - x} + 36) + 12 (\sqrt{2 - x} + 6) - 34$

Add $2$ and $36$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - (38 - x + 6 \sqrt{2 - x} + 6 \sqrt{2 - x}) + 12 (\sqrt{2 - x} + 6) - 34$

Add $6 \sqrt{2 - x}$ and $6 \sqrt{2 - x}$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - (38 - x + 12 \sqrt{2 - x}) + 12 (\sqrt{2 - x} + 6) - 34$

Apply the distributive property.

$f^{- 1} (\sqrt{2 - x} + 6) = - 1 \cdot 38 + x - (12 \sqrt{2 - x}) + 12 (\sqrt{2 - x} + 6) - 34$

Simplify.

Multiply $- 1$ by $38$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - 38 + x - (12 \sqrt{2 - x}) + 12 (\sqrt{2 - x} + 6) - 34$

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - 38 + 1 x - (12 \sqrt{2 - x}) + 12 (\sqrt{2 - x} + 6) - 34$

Multiply $x$ by $1$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - 38 + x - (12 \sqrt{2 - x}) + 12 (\sqrt{2 - x} + 6) - 34$

Multiply $12$ by $- 1$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - 38 + x - 12 \sqrt{2 - x} + 12 (\sqrt{2 - x} + 6) - 34$

Apply the distributive property.

$f^{- 1} (\sqrt{2 - x} + 6) = - 38 + x - 12 \sqrt{2 - x} + 12 \sqrt{2 - x} + 12 \cdot 6 - 34$

Multiply $12$ by $6$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - 38 + x - 12 \sqrt{2 - x} + 12 \sqrt{2 - x} + 72 - 34$

Simplify by adding terms.

Combine the opposite terms in $- 38 + x - 12 \sqrt{2 - x} + 12 \sqrt{2 - x} + 72 - 34$ .

Add $- 12 \sqrt{2 - x}$ and $12 \sqrt{2 - x}$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - 38 + x + 0 + 72 - 34$

Add $- 38 + x$ and $0$ .

$f^{- 1} (\sqrt{2 - x} + 6) = - 38 + x + 72 - 34$

Add $- 38$ and $72$ .

$f^{- 1} (\sqrt{2 - x} + 6) = x + 34 - 34$

Combine the opposite terms in $x + 34 - 34$ .

Subtract $34$ from $34$ .

$f^{- 1} (\sqrt{2 - x} + 6) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\sqrt{2 - x} + 6) = x$

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

Set up the composite result function.

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

Evaluate $f (- x^{2} + 12 x - 34)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (- x^{2} + 12 x - 34) = \sqrt{2 - (- x^{2} + 12 x - 34)} + 6$

Simplify each term.

Apply the distributive property.

$f (- x^{2} + 12 x - 34) = \sqrt{2 + x^{2} - (12 x) + 34} + 6$

Simplify.

Multiply $- - x^{2}$ .

Multiply $- 1$ by $- 1$ .

$f (- x^{2} + 12 x - 34) = \sqrt{2 + 1 x^{2} - (12 x) + 34} + 6$

Multiply $x^{2}$ by $1$ .

$f (- x^{2} + 12 x - 34) = \sqrt{2 + x^{2} - (12 x) + 34} + 6$

Multiply $12$ by $- 1$ .

$f (- x^{2} + 12 x - 34) = \sqrt{2 + x^{2} - 12 x + 34} + 6$

Multiply $- 1$ by $- 34$ .

$f (- x^{2} + 12 x - 34) = \sqrt{2 + x^{2} - 12 x + 34} + 6$

Add $2$ and $34$ .

$f (- x^{2} + 12 x - 34) = \sqrt{x^{2} - 12 x + 36} + 6$

Factor using the perfect square rule.

Rewrite $36$ as $6^{2}$ .

$f (- x^{2} + 12 x - 34) = \sqrt{x^{2} - 12 x + 6^{2}} + 6$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 x = 2 \cdot x \cdot 6$

Rewrite the polynomial.

$f (- x^{2} + 12 x - 34) = \sqrt{x^{2} - 2 \cdot x \cdot 6 + 6^{2}} + 6$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

$f (- x^{2} + 12 x - 34) = \sqrt{{(x - 6)}^{2}} + 6$

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

$f (- x^{2} + 12 x - 34) = x - 6 + 6$

Combine the opposite terms in $x - 6 + 6$ .

Add $- 6$ and $6$ .

$f (- x^{2} + 12 x - 34) = x + 0$

Add $x$ and $0$ .

$f (- x^{2} + 12 x - 34) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = - x^{2} + 12 x - 34$ is the inverse of $f (x) = \sqrt{2 - x} + 6$ .

$f^{- 1} (x) = - x^{2} + 12 x - 34$

Find the Inverse square root of 2-x+6

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