Find the Inverse f(x)=(7-8x)^2

$f (x) = {(7 - 8 x)}^{2}$

Replace $f (x)$ with $y$ .

$y = {(7 - 8 x)}^{2}$

Interchange the variables.

$x = {(7 - 8 y)}^{2}$

Solve for $y$ .

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Rewrite the equation as ${(7 - 8 y)}^{2} = x$ .

${(7 - 8 y)}^{2} = x$

Take the square root of each side of the equation to set up the solution for $y$

${(7 - 8 y)}^{2 \cdot \frac{1}{2}} = \pm \sqrt{x}$

Remove the perfect root factor $7 - 8 y$ under the radical to solve for $y$ .

$7 - 8 y = \pm \sqrt{x}$

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.

$7 - 8 y = \sqrt{x}$

Subtract $7$ from both sides of the equation.

$- 8 y = \sqrt{x} - 7$

Divide each term by $- 8$ and simplify.

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Divide each term in $- 8 y = \sqrt{x} - 7$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{\sqrt{x}}{- 8} + \frac{- 7}{- 8}$

Cancel the common factor of $- 8$ .

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Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{\sqrt{x}}{- 8} + \frac{- 7}{- 8}$

Divide $y$ by $1$ .

$y = \frac{\sqrt{x}}{- 8} + \frac{- 7}{- 8}$

Simplify each term.

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Move the negative in front of the fraction.

$y = - \frac{\sqrt{x}}{8} + \frac{- 7}{- 8}$

Dividing two negative values results in a positive value.

$y = - \frac{\sqrt{x}}{8} + \frac{7}{8}$

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

$7 - 8 y = - \sqrt{x}$

Subtract $7$ from both sides of the equation.

$- 8 y = - \sqrt{x} - 7$

Divide each term by $- 8$ and simplify.

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Divide each term in $- 8 y = - \sqrt{x} - 7$ by $- 8$ .

$\frac{- 8 y}{- 8} = \frac{- \sqrt{x}}{- 8} + \frac{- 7}{- 8}$

Cancel the common factor of $- 8$ .

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Cancel the common factor.

$\frac{- 8 y}{- 8} = \frac{- \sqrt{x}}{- 8} + \frac{- 7}{- 8}$

Divide $y$ by $1$ .

$y = \frac{- \sqrt{x}}{- 8} + \frac{- 7}{- 8}$

Simplify each term.

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Dividing two negative values results in a positive value.

$y = \frac{\sqrt{x}}{8} + \frac{- 7}{- 8}$

Dividing two negative values results in a positive value.

$y = \frac{\sqrt{x}}{8} + \frac{7}{8}$

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

$y = - \frac{\sqrt{x}}{8} + \frac{7}{8}$

$y = \frac{\sqrt{x}}{8} + \frac{7}{8}$

$y = - \frac{\sqrt{x}}{8} + \frac{7}{8}$

$y = \frac{\sqrt{x}}{8} + \frac{7}{8}$

$y = - \frac{\sqrt{x}}{8} + \frac{7}{8}$

$y = \frac{\sqrt{x}}{8} + \frac{7}{8}$

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.

$f^{- 1} (x) = - \frac{\sqrt{x}}{8} + \frac{7}{8}, \frac{\sqrt{x}}{8} + \frac{7}{8}$

Set up the composite result function.

$g (f (x))$

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

$g ({(7 - 8 x)}^{2}) = - \frac{\sqrt{{(7 - 8 x)}^{2}}}{8} + \frac{7}{8}$

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

$g ({(7 - 8 x)}^{2}) = - \frac{7 - 8 x}{8} + \frac{7}{8}$

Combine the numerators over the common denominator.

$g ({(7 - 8 x)}^{2}) = \frac{- (7 - 8 x) + 7}{8}$

Simplify the numerator.

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Apply the distributive property.

$g ({(7 - 8 x)}^{2}) = \frac{- 1 \cdot 7 - (- 8 x) + 7}{8}$

Multiply $- 1$ by $7$ .

$g ({(7 - 8 x)}^{2}) = \frac{- 7 - (- 8 x) + 7}{8}$

Multiply $- 8$ by $- 1$ .

$g ({(7 - 8 x)}^{2}) = \frac{- 7 + 8 x + 7}{8}$

Add $- 7$ and $7$ .

$g ({(7 - 8 x)}^{2}) = \frac{8 x + 0}{8}$

Add $8 x$ and $0$ .

$g ({(7 - 8 x)}^{2}) = \frac{8 x}{8}$

Cancel the common factor of $8$ .

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Cancel the common factor.

$g ({(7 - 8 x)}^{2}) = \frac{8 x}{8}$

Divide $x$ by $1$ .

$g ({(7 - 8 x)}^{2}) = x$

Since $g (f (x)) = x$ , $f^{- 1} (x) = - \frac{\sqrt{x}}{8} + \frac{7}{8}, \frac{\sqrt{x}}{8} + \frac{7}{8}$ is the inverse of $f (x) = {(7 - 8 x)}^{2}$ .

$f^{- 1} (x) = - \frac{\sqrt{x}}{8} + \frac{7}{8}, \frac{\sqrt{x}}{8} + \frac{7}{8}$

Find the Inverse f(x)=(7-8x)^2

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