Find the Inverse y=x^2-18x

$y = x^{2} - 18 x$

Interchange the variables.

$x = y^{2} - 18 y$

Solve for $y$ .

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Rewrite the equation as $y^{2} - 18 y = x$ .

$y^{2} - 18 y = x$

Move $x$ to the left side of the equation by subtracting it from both sides.

$y^{2} - 18 y - x = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = - 18$ , and $c = - x$ into the quadratic formula and solve for $y$ .

$\frac{18 \pm \sqrt{{(- 18)}^{2} - 4 \cdot (1 \cdot (- x))}}{2 \cdot 1}$

Simplify.

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Simplify the numerator.

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Raise $- 18$ to the power of $2$ .

$y = \frac{18 \pm \sqrt{324 - 4 \cdot (1 \cdot (- x))}}{2 \cdot 1}$

Multiply $- x$ by $1$ .

$y = \frac{18 \pm \sqrt{324 - 4 \cdot (- x)}}{2 \cdot 1}$

Multiply $- 1$ by $- 4$ .

$y = \frac{18 \pm \sqrt{324 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $324 + 4 x$ .

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Factor $4$ out of $324$ .

$y = \frac{18 \pm \sqrt{4 \cdot 81 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $4 \cdot 81 + 4 x$ .

$y = \frac{18 \pm \sqrt{4 (81 + x)}}{2 \cdot 1}$

Rewrite $4 (81 + x)$ as $2^{2} (9^{2} + x)$ .

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Rewrite $4$ as $2^{2}$ .

$y = \frac{18 \pm \sqrt{2^{2} (81 + x)}}{2 \cdot 1}$

Rewrite $81$ as $9^{2}$ .

$y = \frac{18 \pm \sqrt{2^{2} (9^{2} + x)}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{18 \pm 2 \sqrt{9^{2} + x}}{2 \cdot 1}$

Raise $9$ to the power of $2$ .

$y = \frac{18 \pm 2 \sqrt{81 + x}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{18 \pm 2 \sqrt{81 + x}}{2}$

Simplify $\frac{18 \pm 2 \sqrt{81 + x}}{2}$ .

$y = 9 \pm \sqrt{81 + x}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $- 18$ to the power of $2$ .

$y = \frac{18 \pm \sqrt{324 - 4 \cdot (1 \cdot (- x))}}{2 \cdot 1}$

Multiply $- x$ by $1$ .

$y = \frac{18 \pm \sqrt{324 - 4 \cdot (- x)}}{2 \cdot 1}$

Multiply $- 1$ by $- 4$ .

$y = \frac{18 \pm \sqrt{324 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $324 + 4 x$ .

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Factor $4$ out of $324$ .

$y = \frac{18 \pm \sqrt{4 \cdot 81 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $4 \cdot 81 + 4 x$ .

$y = \frac{18 \pm \sqrt{4 (81 + x)}}{2 \cdot 1}$

Rewrite $4 (81 + x)$ as $2^{2} (9^{2} + x)$ .

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Rewrite $4$ as $2^{2}$ .

$y = \frac{18 \pm \sqrt{2^{2} (81 + x)}}{2 \cdot 1}$

Rewrite $81$ as $9^{2}$ .

$y = \frac{18 \pm \sqrt{2^{2} (9^{2} + x)}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{18 \pm 2 \sqrt{9^{2} + x}}{2 \cdot 1}$

Raise $9$ to the power of $2$ .

$y = \frac{18 \pm 2 \sqrt{81 + x}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{18 \pm 2 \sqrt{81 + x}}{2}$

Simplify $\frac{18 \pm 2 \sqrt{81 + x}}{2}$ .

$y = 9 \pm \sqrt{81 + x}$

Change the $\pm$ to $+$ .

$y = 9 + \sqrt{81 + x}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $- 18$ to the power of $2$ .

$y = \frac{18 \pm \sqrt{324 - 4 \cdot (1 \cdot (- x))}}{2 \cdot 1}$

Multiply $- x$ by $1$ .

$y = \frac{18 \pm \sqrt{324 - 4 \cdot (- x)}}{2 \cdot 1}$

Multiply $- 1$ by $- 4$ .

$y = \frac{18 \pm \sqrt{324 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $324 + 4 x$ .

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Factor $4$ out of $324$ .

$y = \frac{18 \pm \sqrt{4 \cdot 81 + 4 x}}{2 \cdot 1}$

Factor $4$ out of $4 \cdot 81 + 4 x$ .

$y = \frac{18 \pm \sqrt{4 (81 + x)}}{2 \cdot 1}$

Rewrite $4 (81 + x)$ as $2^{2} (9^{2} + x)$ .

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Rewrite $4$ as $2^{2}$ .

$y = \frac{18 \pm \sqrt{2^{2} (81 + x)}}{2 \cdot 1}$

Rewrite $81$ as $9^{2}$ .

$y = \frac{18 \pm \sqrt{2^{2} (9^{2} + x)}}{2 \cdot 1}$

Pull terms out from under the radical.

$y = \frac{18 \pm 2 \sqrt{9^{2} + x}}{2 \cdot 1}$

Raise $9$ to the power of $2$ .

$y = \frac{18 \pm 2 \sqrt{81 + x}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$y = \frac{18 \pm 2 \sqrt{81 + x}}{2}$

Simplify $\frac{18 \pm 2 \sqrt{81 + x}}{2}$ .

$y = 9 \pm \sqrt{81 + x}$

Change the $\pm$ to $-$ .

$y = 9 - \sqrt{81 + x}$

The final answer is the combination of both solutions.

$y = 9 + \sqrt{81 + x}$

$y = 9 - \sqrt{81 + x}$

$y = 9 + \sqrt{81 + x}$

$y = 9 - \sqrt{81 + x}$

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.

$f^{- 1} (x) = 9 + \sqrt{81 + x}, 9 - \sqrt{81 + x}$

Set up the composite result function.

$g (f (x))$

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

$g (x^{2} - 18 x) = 9 + \sqrt{81 + x^{2} - 18 x}$

Remove parentheses.

$g (x^{2} - 18 x) = 9 + \sqrt{81 + x^{2} - 18 x}$

Simplify each term.

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Factor using the perfect square rule.

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Rearrange terms.

$g (x^{2} - 18 x) = 9 + \sqrt{81 - 18 x + x^{2}}$

Rewrite $81$ as $9^{2}$ .

$g (x^{2} - 18 x) = 9 + \sqrt{9^{2} - 18 x + x^{2}}$

Check the middle term by multiplying $2 a b$ and compare this result with the middle term in the original expression.

$2 a b = 2 \cdot 9 \cdot (- x)$

Simplify.

$2 a b = - 18 x$

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 9$ and $b = - x$ .

$g (x^{2} - 18 x) = 9 + \sqrt{{(9 - x)}^{2}}$

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

$g (x^{2} - 18 x) = 9 + 9 - x$

Add $9$ and $9$ .

$g (x^{2} - 18 x) = 18 - x$

Since $g (f (x)) = x$ , $f^{- 1} (x) = 9 + \sqrt{81 + x}, 9 - \sqrt{81 + x}$ is the inverse of $f (x) = x^{2} - 18 x$ .

$f^{- 1} (x) = 9 + \sqrt{81 + x}, 9 - \sqrt{81 + x}$

Find the Inverse y=x^2-18x

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