Find the Inverse -x^2+6

$- x^{2} + 6$

Interchange the variables.

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

Solve for $y$ .

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

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

Subtract $6$ from both sides of the equation.

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

Multiply each term in $- y^{2} = x - 6$ by $- 1$

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Multiply each term in $- y^{2} = x - 6$ by $- 1$ .

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

Multiply $(- y^{2}) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

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

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

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

Simplify each term.

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Move $- 1$ to the left of $x$ .

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

Rewrite $- 1 x$ as $- x$ .

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

Multiply $- 6$ by $- 1$ .

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

Take the square root of both sides of the equation to eliminate the exponent on the left side.

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

The complete solution is the result of both the positive and negative portions of the solution.

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First, use the positive value of the $\pm$ to find the first solution.

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

Next, use the negative value of the $\pm$ to find the second solution.

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

The complete solution is the result of both the positive and negative portions of the solution.

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

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

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

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

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

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

Solve for $y$ and replace with $f^{- 1} (x)$ .

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

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

Set up the composite result function.

$g (f (x))$

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

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

Apply the distributive property.

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

Multiply $- 1$ by $6$ .

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

Add $- 6$ and $6$ .

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

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

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

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

$g (- x^{2} + 6) = x$

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

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

Find the Inverse -x^2+6

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