Find the Inverse f(x)=5/9(x-32)

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

Step 1

Write $f (x) = \frac{5}{9} (x - 32)$ as an equation.

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

Rewrite the equation as $\frac{5}{9} \cdot (y - 32) = x$ .

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

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

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

Simplify both sides of the equation.

Simplify the left side.

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

Apply the distributive property.

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

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

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

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

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

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

Multiply $5$ by $- 32$ .

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

Move the negative in front of the fraction.

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

Apply the distributive property.

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

Cancel the common factor of $9$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $5$ .

Factor $5$ out of $5 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $9$ .

Move the leading negative in $- \frac{160}{9}$ into the numerator.

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

Cancel the common factor.

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

Rewrite the expression.

$y + \frac{1}{5} \cdot - 160 = \frac{9}{5} x$

Cancel the common factor of $5$ .

Factor $5$ out of $- 160$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

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

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

Add $32$ to both sides of the equation.

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

Step 4

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

$f^{- 1} (x) = \frac{9 x}{5} + 32$

Step 5

Verify if $f^{- 1} (x) = \frac{9 x}{5} + 32$ is the inverse of $f (x) = \frac{5}{9} \cdot (x - 32)$ .

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 $f^{- 1} (\frac{5}{9} \cdot (x - 32))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{5}{9} \cdot (x - 32)) = \frac{9 (\frac{5}{9} \cdot (x - 32))}{5} + 32$

Simplify each term.

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

$f^{- 1} (\frac{5}{9} \cdot (x - 32)) = \frac{\frac{9 \cdot 5}{9} \cdot (x - 32)}{5} + 32$

Multiply $9$ by $5$ .

$f^{- 1} (\frac{5}{9} \cdot (x - 32)) = \frac{\frac{45}{9} \cdot (x - 32)}{5} + 32$

Divide $45$ by $9$ .

$f^{- 1} (\frac{5}{9} \cdot (x - 32)) = \frac{5 (x - 32)}{5} + 32$

Cancel the common factor of $5$ .

Cancel the common factor.

$f^{- 1} (\frac{5}{9} \cdot (x - 32)) = \frac{5 (x - 32)}{5} + 32$

Divide $x - 32$ by $1$ .

$f^{- 1} (\frac{5}{9} \cdot (x - 32)) = x - 32 + 32$

Combine the opposite terms in $x - 32 + 32$ .

Add $- 32$ and $32$ .

$f^{- 1} (\frac{5}{9} \cdot (x - 32)) = x + 0$

Add $x$ and $0$ .

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

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

Set up the composite result function.

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

Evaluate $f (\frac{9 x}{5} + 32)$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Combine the opposite terms in $(\frac{9 x}{5} + 32) - 32$ .

Subtract $32$ from $32$ .

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

Add $\frac{9 x}{5}$ and $0$ .

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

$f (\frac{9 x}{5} + 32) = \frac{1}{9} \cdot (9 x)$

Cancel the common factor of $9$ .

Factor $9$ out of $9 x$ .

$f (\frac{9 x}{5} + 32) = \frac{1}{9} \cdot (9 (x))$

Cancel the common factor.

$f (\frac{9 x}{5} + 32) = \frac{1}{9} \cdot (9 x)$

Rewrite the expression.

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

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

$f^{- 1} (x) = \frac{9 x}{5} + 32$

Find the Inverse f(x)=5/9(x-32)

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