Find the Inverse 9/5x+32

$\frac{9}{5} x + 32$

Interchange the variables.

$x = \frac{9}{5} y + 32$

Solve for $y$ .

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Rewrite the equation as $\frac{9}{5} y + 32 = x$ .

$\frac{9}{5} y + 32 = x$

Combine $\frac{9}{5}$ and $y$ .

$\frac{9 y}{5} + 32 = x$

Subtract $32$ from both sides of the equation.

$\frac{9 y}{5} = x - 32$

Multiply both sides of the equation by $\frac{5}{9}$ .

$\frac{5}{9} \cdot \frac{9 y}{5} = \frac{5}{9} \cdot (x - 32)$

Simplify both sides of the equation.

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Simplify $\frac{5}{9} \cdot \frac{9 y}{5}$ .

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Cancel the common factor of $5$ .

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Cancel the common factor.

$\frac{5}{9} \cdot \frac{9 y}{5} = \frac{5}{9} \cdot (x - 32)$

Rewrite the expression.

$\frac{1}{9} \cdot (9 y) = \frac{5}{9} \cdot (x - 32)$

Cancel the common factor of $9$ .

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Factor $9$ out of $9 y$ .

$\frac{1}{9} \cdot (9 (y)) = \frac{5}{9} \cdot (x - 32)$

Cancel the common factor.

$\frac{1}{9} \cdot (9 y) = \frac{5}{9} \cdot (x - 32)$

Rewrite the expression.

$y = \frac{5}{9} \cdot (x - 32)$

Simplify $\frac{5}{9} \cdot (x - 32)$ .

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Apply the distributive property.

$y = \frac{5}{9} x + \frac{5}{9} \cdot - 32$

Combine $\frac{5}{9}$ and $x$ .

$y = \frac{5 x}{9} + \frac{5}{9} \cdot - 32$

Multiply $\frac{5}{9} \cdot - 32$ .

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Combine $\frac{5}{9}$ and $- 32$ .

$y = \frac{5 x}{9} + \frac{5 \cdot - 32}{9}$

Multiply $5$ by $- 32$ .

$y = \frac{5 x}{9} + \frac{- 160}{9}$

Move the negative in front of the fraction.

$y = \frac{5 x}{9} - \frac{160}{9}$

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.

$f^{- 1} (x) = \frac{5 x}{9} - \frac{160}{9}$

Set up the composite result function.

$g (f (x))$

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

$g (\frac{9}{5} x + 32) = \frac{5 (\frac{9}{5} x + 32)}{9} - \frac{160}{9}$

Simplify terms.

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Simplify each term.

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Simplify the numerator.

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Combine $\frac{9}{5}$ and $x$ .

$g (\frac{9}{5} x + 32) = \frac{5 (\frac{9 x}{5} + 32)}{9} - \frac{160}{9}$

To write $32$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$g (\frac{9}{5} x + 32) = \frac{5 (\frac{9 x}{5} + 32 \cdot \frac{5}{5})}{9} - \frac{160}{9}$

Combine $32$ and $\frac{5}{5}$ .

$g (\frac{9}{5} x + 32) = \frac{5 (\frac{9 x}{5} + \frac{32 \cdot 5}{5})}{9} - \frac{160}{9}$

Combine the numerators over the common denominator.

$g (\frac{9}{5} x + 32) = \frac{5 (\frac{9 x + 32 \cdot 5}{5})}{9} - \frac{160}{9}$

Multiply $32$ by $5$ .

$g (\frac{9}{5} x + 32) = \frac{5 (\frac{9 x + 160}{5})}{9} - \frac{160}{9}$

Combine $5$ and $\frac{9 x + 160}{5}$ .

$g (\frac{9}{5} x + 32) = \frac{\frac{5 (9 x + 160)}{5}}{9} - \frac{160}{9}$

Reduce the expression by cancelling the common factors.

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Reduce the expression $\frac{5 (9 x + 160)}{5}$ by cancelling the common factors.

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Cancel the common factor.

$g (\frac{9}{5} x + 32) = \frac{\frac{5 (9 x + 160)}{5}}{9} - \frac{160}{9}$

Rewrite the expression.

$g (\frac{9}{5} x + 32) = \frac{\frac{9 x + 160}{1}}{9} - \frac{160}{9}$

Divide $9 x + 160$ by $1$ .

$g (\frac{9}{5} x + 32) = \frac{9 x + 160}{9} - \frac{160}{9}$

Combine the numerators over the common denominator.

$g (\frac{9}{5} x + 32) = \frac{9 x + 160 - 160}{9}$

Simplify the numerator.

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Subtract $160$ from $160$ .

$g (\frac{9}{5} x + 32) = \frac{9 x + 0}{9}$

Add $9 x$ and $0$ .

$g (\frac{9}{5} x + 32) = \frac{9 x}{9}$

Cancel the common factor of $9$ .

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Cancel the common factor.

$g (\frac{9}{5} x + 32) = \frac{9 x}{9}$

Divide $x$ by $1$ .

$g (\frac{9}{5} x + 32) = x$

Since $g (f (x)) = x$ , $f^{- 1} (x) = \frac{5 x}{9} - \frac{160}{9}$ is the inverse of $f (x) = \frac{9}{5} x + 32$ .

$f^{- 1} (x) = \frac{5 x}{9} - \frac{160}{9}$

Find the Inverse 9/5x+32

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