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

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

Step 1

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

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

Step 2

Interchange the variables.

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

Step 3

Solve for $y$ .

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

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

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

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

Simplify both sides of the equation.

Simplify the left side.

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

Apply the distributive property.

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

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

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

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

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

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

Multiply $5$ by $- 32$ .

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

Move the negative in front of the fraction.

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

Apply the distributive property.

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

Cancel the common factor of $9$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $5$ .

Factor $5$ out of $5 y$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $9$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $5$ .

Factor $5$ out of $- 160$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

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

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

Add $32$ to both sides of the equation.

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

Step 4

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

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

Step 5

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

To verify the inverse, check if $C^{- 1} (C (f)) = f$ and $C (C^{- 1} (f)) = f$ .

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

Set up the composite result function.

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

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

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

Simplify each term.

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

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

Multiply $9$ by $5$ .

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

Divide $45$ by $9$ .

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Divide $f - 32$ by $1$ .

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

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

Add $- 32$ and $32$ .

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

Add $f$ and $0$ .

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

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

Set up the composite result function.

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

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

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

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

Subtract $32$ from $32$ .

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $9$ .

Factor $9$ out of $9 f$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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