Precalculus Examples

$f (x) = - \frac{4}{5} x + 8$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$f (x + h) = - \frac{4}{5} \cdot (x + h) + 8$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Apply the distributive property.

$f (x + h) = - \frac{4}{5} x - \frac{4}{5} h + 8$

Step 2.1.2.1.2

Combine $x$ and $\frac{4}{5}$ .

$f (x + h) = - \frac{x \cdot 4}{5} - \frac{4}{5} h + 8$

Step 2.1.2.1.3

Combine $h$ and $\frac{4}{5}$ .

$f (x + h) = - \frac{x \cdot 4}{5} - \frac{h \cdot 4}{5} + 8$

Step 2.1.2.1.4

Simplify each term.

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Step 2.1.2.1.4.1

Move $4$ to the left of $x$ .

$f (x + h) = - \frac{4 \cdot x}{5} - \frac{h \cdot 4}{5} + 8$

Step 2.1.2.1.4.2

Move $4$ to the left of $h$ .

$f (x + h) = - \frac{4 x}{5} - \frac{4 h}{5} + 8$

Step 2.1.2.2

The final answer is $- \frac{4 x}{5} - \frac{4 h}{5} + 8$ .

$- \frac{4 x}{5} - \frac{4 h}{5} + 8$

Step 2.2

Reorder $- \frac{4 x}{5}$ and $- \frac{4 h}{5}$ .

$- \frac{4 h}{5} - \frac{4 x}{5} + 8$

Step 2.3

Find the components of the definition.

$f (x + h) = - \frac{4 h}{5} - \frac{4 x}{5} + 8$

$f (x) = - \frac{4 x}{5} + 8$

$f (x + h) = - \frac{4 h}{5} - \frac{4 x}{5} + 8$

$f (x) = - \frac{4 x}{5} + 8$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- \frac{4 h}{5} - \frac{4 x}{5} + 8 - (- \frac{4 x}{5} + 8)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

$\frac{- \frac{4 h}{5} - \frac{4 x}{5} + 8 - - \frac{4 x}{5} - 1 \cdot 8}{h}$

Step 4.1.2

Multiply $- - \frac{4 x}{5}$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{- \frac{4 h}{5} - \frac{4 x}{5} + 8 + 1 \frac{4 x}{5} - 1 \cdot 8}{h}$

Step 4.1.2.2

Multiply $\frac{4 x}{5}$ by $1$ .

$\frac{- \frac{4 h}{5} - \frac{4 x}{5} + 8 + \frac{4 x}{5} - 1 \cdot 8}{h}$

Step 4.1.3

Multiply $- 1$ by $8$ .

$\frac{- \frac{4 h}{5} - \frac{4 x}{5} + 8 + \frac{4 x}{5} - 8}{h}$

Step 4.1.4

Add $- \frac{4 x}{5}$ and $\frac{4 x}{5}$ .

$\frac{- \frac{4 h}{5} + 0 + 8 - 8}{h}$

Step 4.1.5

Add $- \frac{4 h}{5}$ and $0$ .

$\frac{- \frac{4 h}{5} + 8 - 8}{h}$

Step 4.1.6

Subtract $8$ from $8$ .

$\frac{- \frac{4 h}{5} + 0}{h}$

Step 4.1.7

Add $- \frac{4 h}{5}$ and $0$ .

$\frac{- \frac{4 h}{5}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{4 h}{5} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{4 h}{5}$ into the numerator.

$\frac{- 4 h}{5} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- 4 h$ .

$\frac{h \cdot - 4}{5} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 4}{5} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 4}{5}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{4}{5}$

Step 5