Precalculus Examples

$f (x) = 0.5 (- 10 x + 24)$

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.

$f (x + h) = 0.5 (- 10 (x + h) + 24)$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = 0.5 (- 10 x - 10 h + 24)$

Step 2.1.2.2

Apply the distributive property.

$f (x + h) = 0.5 (- 10 x) + 0.5 (- 10 h) + 0.5 \cdot 24$

Step 2.1.2.3

Simplify.

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Step 2.1.2.3.1

Multiply $- 10$ by $0.5$ .

$f (x + h) = - 5 x + 0.5 (- 10 h) + 0.5 \cdot 24$

Step 2.1.2.3.2

Multiply $- 10$ by $0.5$ .

$f (x + h) = - 5 x - 5 h + 0.5 \cdot 24$

Step 2.1.2.3.3

Multiply $0.5$ by $24$ .

$f (x + h) = - 5 x - 5 h + 12$

Step 2.1.2.4

The final answer is $- 5 x - 5 h + 12$ .

$- 5 x - 5 h + 12$

Step 2.2

Reorder $- 5 x$ and $- 5 h$ .

$- 5 h - 5 x + 12$

Step 2.3

Find the components of the definition.

$f (x + h) = - 5 h - 5 x + 12$

$f (x) = - 5 x + 12$

$f (x + h) = - 5 h - 5 x + 12$

$f (x) = - 5 x + 12$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 5 h - 5 x + 12 - (- 5 x + 12)}{h}$

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.

$\frac{- 5 h - 5 x + 12 - (- 5 x) - 1 \cdot 12}{h}$

Step 4.1.2

Multiply $- 5$ by $- 1$ .

$\frac{- 5 h - 5 x + 12 + 5 x - 1 \cdot 12}{h}$

Step 4.1.3

Multiply $- 1$ by $12$ .

$\frac{- 5 h - 5 x + 12 + 5 x - 12}{h}$

Step 4.1.4

Add $- 5 x$ and $5 x$ .

$\frac{- 5 h + 0 x + 12 - 12}{h}$

Step 4.1.5

Subtract $12$ from $12$ .

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

Step 4.1.6

Add $- 5 h + 0 x$ and $0$ .

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

Step 4.1.7

Rewrite $- 5 h + 0 x$ in a factored form.

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Step 4.1.7.1

Factor $5$ out of $- 5 h + 0 x$ .

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Step 4.1.7.1.1

Factor $5$ out of $- 5 h$ .

$\frac{5 (- h) + 0 x}{h}$

Step 4.1.7.1.2

Factor $5$ out of $0 x$ .

$\frac{5 (- h) + 5 (0 x)}{h}$

Step 4.1.7.1.3

Factor $5$ out of $5 (- h) + 5 (0 x)$ .

$\frac{5 (- h + 0 x)}{h}$

Step 4.1.7.2

Multiply $0$ by $x$ .

$\frac{5 (- h + 0)}{h}$

Step 4.1.7.3

Add $- h$ and $0$ .

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

Step 4.1.7.4

Combine exponents.

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Step 4.1.7.4.1

Factor out negative.

$\frac{- (5 h)}{h}$

Step 4.1.7.4.2

Multiply $5$ by $- 1$ .

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

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Divide $- 5$ by $1$ .

$- 5$

Step 5