Precalculus Examples

$y = 5.45 + 1.05 (x + 7)$

Step 1

Write $y = 5.45 + 1.05 (x + 7)$ as a function.

$f (x) = 5.45 + 1.05 (x + 7)$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (x + h) = 5.45 + 1.05 ((x + h) + 7)$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Apply the distributive property.

$f (x + h) = 5.45 + 1.05 x + 1.05 h + 1.05 \cdot 7$

Step 3.1.2.1.2

Multiply $1.05$ by $7$ .

$f (x + h) = 5.45 + 1.05 x + 1.05 h + 7.35$

Step 3.1.2.2

Add $5.45$ and $7.35$ .

$f (x + h) = 1.05 x + 1.05 h + 12.8$

Step 3.1.2.3

The final answer is $1.05 x + 1.05 h + 12.8$ .

$1.05 x + 1.05 h + 12.8$

Step 3.2

Reorder $1.05 x$ and $1.05 h$ .

$1.05 h + 1.05 x + 12.8$

Step 3.3

Find the components of the definition.

$f (x + h) = 1.05 h + 1.05 x + 12.8$

$f (x) = 1.05 x + 12.8$

$f (x + h) = 1.05 h + 1.05 x + 12.8$

$f (x) = 1.05 x + 12.8$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{1.05 h + 1.05 x + 12.8 - (1.05 x + 12.8)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $0.05$ out of $1.05 h + 1.05 x + 12.8 - (1.05 x + 12.8)$ .

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Step 5.1.1.1

Factor $0.05$ out of $1.05 h$ .

$\frac{0.05 (21 h) + 1.05 x + 12.8 - (1.05 x + 12.8)}{h}$

Step 5.1.1.2

Factor $0.05$ out of $1.05 x$ .

$\frac{0.05 (21 h) + 0.05 (21 x) + 12.8 - (1.05 x + 12.8)}{h}$

Step 5.1.1.3

Factor $0.05$ out of $12.8$ .

$\frac{0.05 (21 h) + 0.05 (21 x) + 0.05 (256) - (1.05 x + 12.8)}{h}$

Step 5.1.1.4

Factor $0.05$ out of $- (1.05 x + 12.8)$ .

$\frac{0.05 (21 h) + 0.05 (21 x) + 0.05 (256) + 0.05 (- 20 (1.05 x + 12.8))}{h}$

Step 5.1.1.5

Factor $0.05$ out of $0.05 (21 h) + 0.05 (21 x)$ .

$\frac{0.05 (21 h + 21 x) + 0.05 (256) + 0.05 (- 20 (1.05 x + 12.8))}{h}$

Step 5.1.1.6

Factor $0.05$ out of $0.05 (21 h + 21 x) + 0.05 (256)$ .

$\frac{0.05 (21 h + 21 x + 256) + 0.05 (- 20 (1.05 x + 12.8))}{h}$

Step 5.1.1.7

Factor $0.05$ out of $0.05 (21 h + 21 x + 256) + 0.05 (- 20 (1.05 x + 12.8))$ .

$\frac{0.05 (21 h + 21 x + 256 - 20 (1.05 x + 12.8))}{h}$

Step 5.1.2

Apply the distributive property.

$\frac{0.05 (21 h + 21 x + 256 - 20 (1.05 x) - 20 \cdot 12.8)}{h}$

Step 5.1.3

Multiply $1.05$ by $- 20$ .

$\frac{0.05 (21 h + 21 x + 256 - 21 x - 20 \cdot 12.8)}{h}$

Step 5.1.4

Multiply $- 20$ by $12.8$ .

$\frac{0.05 (21 h + 21 x + 256 - 21 x - 256)}{h}$

Step 5.1.5

Subtract $21 x$ from $21 x$ .

$\frac{0.05 (21 h + 0 x + 256 - 256)}{h}$

Step 5.1.6

Subtract $256$ from $256$ .

$\frac{0.05 (21 h + 0 x + 0)}{h}$

Step 5.1.7

Add $21 h + 0 x$ and $0$ .

$\frac{0.05 (21 h + 0 x)}{h}$

Step 5.1.8

Rewrite $21 h + 0 x$ in a factored form.

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Step 5.1.8.1

Factor $21$ out of $21 h + 0 x$ .

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Step 5.1.8.1.1

Factor $21$ out of $21 h$ .

$\frac{0.05 (21 h + 0 x)}{h}$

Step 5.1.8.1.2

Factor $21$ out of $0 x$ .

$\frac{0.05 (21 h + 21 (0 x))}{h}$

Step 5.1.8.1.3

Factor $21$ out of $21 h + 21 (0 x)$ .

$\frac{0.05 (21 (h + 0 x))}{h}$

Step 5.1.8.2

Multiply $0$ by $x$ .

$\frac{0.05 (21 (h + 0))}{h}$

Step 5.1.8.3

Add $h$ and $0$ .

$\frac{0.05 (21 h)}{h}$

$\frac{0.05 \cdot 21 h}{h}$

Step 5.1.9

Multiply $0.05$ by $21$ .

$\frac{1.05 h}{h}$

Step 5.2

Cancel the common factor of $h$ .

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Step 5.2.1

Cancel the common factor.

$\frac{1.05 h}{h}$

Step 5.2.2

Divide $1.05$ by $1$ .

$1.05$

Step 6