Precalculus Examples

$f (x) = - 2 x + 4$

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.

$f (x + h) = - 2 (x + h) + 4$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

$f (x + h) = - 2 x - 2 h + 4$

Step 2.1.2.2

The final answer is $- 2 x - 2 h + 4$ .

$- 2 x - 2 h + 4$

Step 2.2

Reorder $- 2 x$ and $- 2 h$ .

$- 2 h - 2 x + 4$

Step 2.3

Find the components of the definition.

$f (x + h) = - 2 h - 2 x + 4$

$f (x) = - 2 x + 4$

$f (x + h) = - 2 h - 2 x + 4$

$f (x) = - 2 x + 4$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{- 2 h - 2 x + 4 - (- 2 x + 4)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Factor $2$ out of $- 2 x + 4$ .

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Step 4.1.1.1

Factor $2$ out of $- 2 x$ .

$\frac{- 2 h - 2 x + 4 - (2 (- x) + 4)}{h}$

Step 4.1.1.2

Factor $2$ out of $4$ .

$\frac{- 2 h - 2 x + 4 - (2 (- x) + 2 (2))}{h}$

Step 4.1.1.3

Factor $2$ out of $2 (- x) + 2 (2)$ .

$\frac{- 2 h - 2 x + 4 - (2 (- x + 2))}{h}$

$\frac{- 2 h - 2 x + 4 - 1 \cdot 2 (- x + 2)}{h}$

Step 4.1.2

Multiply $- 1$ by $2$ .

$\frac{- 2 h - 2 x + 4 - 2 (- x + 2)}{h}$

Step 4.1.3

Factor $2$ out of $- 2 h - 2 x + 4 - 2 (- x + 2)$ .

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Step 4.1.3.1

Factor $2$ out of $- 2 h$ .

$\frac{2 (- h) - 2 x + 4 - 2 (- x + 2)}{h}$

Step 4.1.3.2

Factor $2$ out of $- 2 x$ .

$\frac{2 (- h) + 2 (- x) + 4 - 2 (- x + 2)}{h}$

Step 4.1.3.3

Factor $2$ out of $4$ .

$\frac{2 (- h) + 2 (- x) + 2 (2) - 2 (- x + 2)}{h}$

Step 4.1.3.4

Factor $2$ out of $- 2 (- x + 2)$ .

$\frac{2 (- h) + 2 (- x) + 2 (2) + 2 (- (- x + 2))}{h}$

Step 4.1.3.5

Factor $2$ out of $2 (- h) + 2 (- x)$ .

$\frac{2 (- h - x) + 2 (2) + 2 (- (- x + 2))}{h}$

Step 4.1.3.6

Factor $2$ out of $2 (- h - x) + 2 (2)$ .

$\frac{2 (- h - x + 2) + 2 (- (- x + 2))}{h}$

Step 4.1.3.7

Factor $2$ out of $2 (- h - x + 2) + 2 (- (- x + 2))$ .

$\frac{2 (- h - x + 2 - (- x + 2))}{h}$

Step 4.1.4

Apply the distributive property.

$\frac{2 (- h - x + 2 - - x - 1 \cdot 2)}{h}$

Step 4.1.5

Multiply $- - x$ .

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Step 4.1.5.1

Multiply $- 1$ by $- 1$ .

$\frac{2 (- h - x + 2 + 1 x - 1 \cdot 2)}{h}$

Step 4.1.5.2

Multiply $x$ by $1$ .

$\frac{2 (- h - x + 2 + x - 1 \cdot 2)}{h}$

Step 4.1.6

Multiply $- 1$ by $2$ .

$\frac{2 (- h - x + 2 + x - 2)}{h}$

Step 4.1.7

Add $- x$ and $x$ .

$\frac{2 (- h + 0 + 2 - 2)}{h}$

Step 4.1.8

Add $- h$ and $0$ .

$\frac{2 (- h + 2 - 2)}{h}$

Step 4.1.9

Subtract $2$ from $2$ .

$\frac{2 (- h + 0)}{h}$

Step 4.1.10

Add $- h$ and $0$ .

$\frac{2 \cdot - 1 h}{h}$

Step 4.1.11

Combine exponents.

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Step 4.1.11.1

Factor out negative.

$\frac{- (2 h)}{h}$

Step 4.1.11.2

Multiply $2$ by $- 1$ .

$\frac{- 2 h}{h}$

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

$\frac{- 2 h}{h}$

Step 4.2.2

Divide $- 2$ by $1$ .

$- 2$

Step 5