Algebra Examples

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

Step 1

Interchange the variables.

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Subtract $32$ from both sides of the equation.

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

Step 2.4

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

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

Step 2.5

Simplify both sides of the equation.

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Step 2.5.1

Simplify the left side.

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Step 2.5.1.1

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

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Step 2.5.1.1.1

Cancel the common factor of $5$ .

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Step 2.5.1.1.1.1

Cancel the common factor.

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

Step 2.5.1.1.1.2

Rewrite the expression.

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

Step 2.5.1.1.2

Cancel the common factor of $9$ .

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Step 2.5.1.1.2.1

Factor $9$ out of $9 y$ .

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

Step 2.5.1.1.2.2

Cancel the common factor.

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

Step 2.5.1.1.2.3

Rewrite the expression.

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

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

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

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Step 2.5.2.1.1

Apply the distributive property.

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

Step 2.5.2.1.2

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

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

Step 2.5.2.1.3

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

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Step 2.5.2.1.3.1

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

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

Step 2.5.2.1.3.2

Multiply $5$ by $- 32$ .

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

Step 2.5.2.1.4

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

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

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Step 4.1

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

Step 4.2

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

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Step 4.2.1

Set up the composite result function.

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

Step 4.2.2

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

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

Step 4.2.3

Combine the numerators over the common denominator.

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

Step 4.2.4

Simplify each term.

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Step 4.2.4.1

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

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

Step 4.2.4.2

Apply the distributive property.

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

Step 4.2.4.3

Cancel the common factor of $5$ .

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Step 4.2.4.3.1

Cancel the common factor.

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

Step 4.2.4.3.2

Rewrite the expression.

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

Step 4.2.4.4

Multiply $5$ by $32$ .

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

Step 4.2.5

Simplify terms.

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Step 4.2.5.1

Combine the opposite terms in $9 x + 160 - 160$ .

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Step 4.2.5.1.1

Subtract $160$ from $160$ .

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

Step 4.2.5.1.2

Add $9 x$ and $0$ .

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

Step 4.2.5.2

Cancel the common factor of $9$ .

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Step 4.2.5.2.1

Cancel the common factor.

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

Step 4.2.5.2.2

Divide $x$ by $1$ .

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

Step 4.3

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

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Step 4.3.1

Set up the composite result function.

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

Step 4.3.2

Evaluate $f (\frac{5 x}{9} - \frac{160}{9})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Simplify each term.

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Step 4.3.3.1

Apply the distributive property.

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

Step 4.3.3.2

Cancel the common factor of $9$ .

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Step 4.3.3.2.1

Cancel the common factor.

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

Step 4.3.3.2.2

Rewrite the expression.

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

Step 4.3.3.3

Cancel the common factor of $5$ .

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Step 4.3.3.3.1

Factor $5$ out of $5 x$ .

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

Step 4.3.3.3.2

Cancel the common factor.

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

Step 4.3.3.3.3

Rewrite the expression.

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

Step 4.3.3.4

Cancel the common factor of $9$ .

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Step 4.3.3.4.1

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

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

Step 4.3.3.4.2

Cancel the common factor.

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

Step 4.3.3.4.3

Rewrite the expression.

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

Step 4.3.3.5

Cancel the common factor of $5$ .

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Step 4.3.3.5.1

Factor $5$ out of $- 160$ .

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

Step 4.3.3.5.2

Cancel the common factor.

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

Step 4.3.3.5.3

Rewrite the expression.

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

Step 4.3.4

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

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Step 4.3.4.1

Add $- 32$ and $32$ .

$f (\frac{5 x}{9} - \frac{160}{9}) = x + 0$

Step 4.3.4.2

Add $x$ and $0$ .

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

Step 4.4

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

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