Precalculus Examples

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

Step 1

Write $C = \frac{5}{9} (F - 32)$ as a function.

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

Step 2

Consider the difference quotient formula.

$\frac{f (F + h) - f F}{h}$

Step 3

Find the components of the definition.

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Step 3.1

Evaluate the function at $x = F + h$ .

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Step 3.1.1

Replace the variable $F$ with $F + h$ in the expression.

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

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

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

Step 3.1.2.2

Simplify.

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Step 3.1.2.2.1

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

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

Step 3.1.2.2.2

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

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

Step 3.1.2.2.3

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

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Step 3.1.2.2.3.1

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

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

Step 3.1.2.2.3.2

Multiply $5$ by $- 32$ .

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

Step 3.1.2.3

Move the negative in front of the fraction.

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

Step 3.1.2.4

The final answer is $\frac{5 F}{9} + \frac{5 h}{9} - \frac{160}{9}$ .

$\frac{5 F}{9} + \frac{5 h}{9} - \frac{160}{9}$

Step 3.2

Find the components of the definition.

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

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

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

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

Step 4

Plug in the components.

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

$\frac{\frac{5 F}{9} + \frac{5 h}{9} - \frac{160}{9} - \frac{5 F}{9} - - \frac{160}{9}}{h}$

Step 5.1.2

Multiply $- - \frac{160}{9}$ .

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Step 5.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{\frac{5 F}{9} + \frac{5 h}{9} - \frac{160}{9} - \frac{5 F}{9} + 1 (\frac{160}{9})}{h}$

Step 5.1.2.2

Multiply $\frac{160}{9}$ by $1$ .

$\frac{\frac{5 F}{9} + \frac{5 h}{9} - \frac{160}{9} - \frac{5 F}{9} + \frac{160}{9}}{h}$

Step 5.1.3

Subtract $\frac{5 F}{9}$ from $\frac{5 F}{9}$ .

$\frac{0 + \frac{5 h}{9} - \frac{160}{9} + \frac{160}{9}}{h}$

Step 5.1.4

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

$\frac{\frac{5 h}{9} - \frac{160}{9} + \frac{160}{9}}{h}$

Step 5.1.5

Add $- \frac{160}{9}$ and $\frac{160}{9}$ .

$\frac{\frac{5 h}{9} + 0}{h}$

Step 5.1.6

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

$\frac{\frac{5 h}{9}}{h}$

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{5 h}{9} \cdot \frac{1}{h}$

Step 5.3

Cancel the common factor of $h$ .

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Step 5.3.1

Factor $h$ out of $5 h$ .

$\frac{h \cdot 5}{9} \cdot \frac{1}{h}$

Step 5.3.2

Cancel the common factor.

$\frac{h \cdot 5}{9} \cdot \frac{1}{h}$

Step 5.3.3

Rewrite the expression.

$\frac{5}{9}$

Step 6