Precalculus Examples

$y = 3.75 + 1.5 (x - 1)$

Step 1

Write $y = 3.75 + 1.5 (x - 1)$ as a function.

$f (x) = 3.75 + 1.5 (x - 1)$

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.

$f (x + h) = 3.75 + 1.5 ((x + h) - 1)$

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.

$f (x + h) = 3.75 + 1.5 x + 1.5 h + 1.5 \cdot - 1$

Step 3.1.2.1.2

Multiply $1.5$ by $- 1$ .

$f (x + h) = 3.75 + 1.5 x + 1.5 h - 1.5$

Step 3.1.2.2

Subtract $1.5$ from $3.75$ .

$f (x + h) = 1.5 x + 1.5 h + 2.25$

Step 3.1.2.3

The final answer is $1.5 x + 1.5 h + 2.25$ .

$1.5 x + 1.5 h + 2.25$

Step 3.2

Reorder $1.5 x$ and $1.5 h$ .

$1.5 h + 1.5 x + 2.25$

Step 3.3

Find the components of the definition.

$f (x + h) = 1.5 h + 1.5 x + 2.25$

$f (x) = 1.5 x + 2.25$

$f (x + h) = 1.5 h + 1.5 x + 2.25$

$f (x) = 1.5 x + 2.25$

Step 4

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{1.5 h + 1.5 x + 2.25 - (1.5 x + 2.25)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $0.25$ out of $1.5 h + 1.5 x + 2.25 - (1.5 x + 2.25)$ .

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Step 5.1.1.1

Factor $0.25$ out of $1.5 h$ .

$\frac{0.25 (6 h) + 1.5 x + 2.25 - (1.5 x + 2.25)}{h}$

Step 5.1.1.2

Factor $0.25$ out of $1.5 x$ .

$\frac{0.25 (6 h) + 0.25 (6 x) + 2.25 - (1.5 x + 2.25)}{h}$

Step 5.1.1.3

Factor $0.25$ out of $2.25$ .

$\frac{0.25 (6 h) + 0.25 (6 x) + 0.25 (9) - (1.5 x + 2.25)}{h}$

Step 5.1.1.4

Factor $0.25$ out of $- (1.5 x + 2.25)$ .

$\frac{0.25 (6 h) + 0.25 (6 x) + 0.25 (9) + 0.25 (- 4 (1.5 x + 2.25))}{h}$

Step 5.1.1.5

Factor $0.25$ out of $0.25 (6 h) + 0.25 (6 x)$ .

$\frac{0.25 (6 h + 6 x) + 0.25 (9) + 0.25 (- 4 (1.5 x + 2.25))}{h}$

Step 5.1.1.6

Factor $0.25$ out of $0.25 (6 h + 6 x) + 0.25 (9)$ .

$\frac{0.25 (6 h + 6 x + 9) + 0.25 (- 4 (1.5 x + 2.25))}{h}$

Step 5.1.1.7

Factor $0.25$ out of $0.25 (6 h + 6 x + 9) + 0.25 (- 4 (1.5 x + 2.25))$ .

$\frac{0.25 (6 h + 6 x + 9 - 4 (1.5 x + 2.25))}{h}$

Step 5.1.2

Factor $0.75$ out of $1.5 x + 2.25$ .

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Step 5.1.2.1

Factor $0.75$ out of $1.5 x$ .

$\frac{0.25 (6 h + 6 x + 9 - 4 (0.75 (2 x) + 2.25))}{h}$

Step 5.1.2.2

Factor $0.75$ out of $2.25$ .

$\frac{0.25 (6 h + 6 x + 9 - 4 (0.75 (2 x) + 0.75 (3)))}{h}$

Step 5.1.2.3

Factor $0.75$ out of $0.75 (2 x) + 0.75 (3)$ .

$\frac{0.25 (6 h + 6 x + 9 - 4 (0.75 (2 x + 3)))}{h}$

$\frac{0.25 (6 h + 6 x + 9 - 4 \cdot 0.75 (2 x + 3))}{h}$

Step 5.1.3

Multiply $- 4$ by $0.75$ .

$\frac{0.25 (6 h + 6 x + 9 - 3 (2 x + 3))}{h}$

Step 5.1.4

Factor $3$ out of $6 h + 6 x + 9 - 3 (2 x + 3)$ .

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Step 5.1.4.1

Factor $3$ out of $6 h$ .

$\frac{0.25 (3 (2 h) + 6 x + 9 - 3 (2 x + 3))}{h}$

Step 5.1.4.2

Factor $3$ out of $6 x$ .

$\frac{0.25 (3 (2 h) + 3 (2 x) + 9 - 3 (2 x + 3))}{h}$

Step 5.1.4.3

Factor $3$ out of $9$ .

$\frac{0.25 (3 (2 h) + 3 (2 x) + 3 (3) - 3 (2 x + 3))}{h}$

Step 5.1.4.4

Factor $3$ out of $- 3 (2 x + 3)$ .

$\frac{0.25 (3 (2 h) + 3 (2 x) + 3 (3) + 3 (- (2 x + 3)))}{h}$

Step 5.1.4.5

Factor $3$ out of $3 (2 h) + 3 (2 x)$ .

$\frac{0.25 (3 (2 h + 2 x) + 3 (3) + 3 (- (2 x + 3)))}{h}$

Step 5.1.4.6

Factor $3$ out of $3 (2 h + 2 x) + 3 (3)$ .

$\frac{0.25 (3 (2 h + 2 x + 3) + 3 (- (2 x + 3)))}{h}$

Step 5.1.4.7

Factor $3$ out of $3 (2 h + 2 x + 3) + 3 (- (2 x + 3))$ .

$\frac{0.25 (3 (2 h + 2 x + 3 - (2 x + 3)))}{h}$

Step 5.1.5

Apply the distributive property.

$\frac{0.25 (3 (2 h + 2 x + 3 - (2 x) - 1 \cdot 3))}{h}$

Step 5.1.6

Multiply $2$ by $- 1$ .

$\frac{0.25 (3 (2 h + 2 x + 3 - 2 x - 1 \cdot 3))}{h}$

Step 5.1.7

Multiply $- 1$ by $3$ .

$\frac{0.25 (3 (2 h + 2 x + 3 - 2 x - 3))}{h}$

Step 5.1.8

Subtract $2 x$ from $2 x$ .

$\frac{0.25 (3 (2 h + 0 + 3 - 3))}{h}$

Step 5.1.9

Add $2 h$ and $0$ .

$\frac{0.25 (3 (2 h + 3 - 3))}{h}$

Step 5.1.10

Subtract $3$ from $3$ .

$\frac{0.25 (3 (2 h + 0))}{h}$

Step 5.1.11

Add $2 h$ and $0$ .

$\frac{0.25 (3 \cdot 2 h)}{h}$

Step 5.1.12

Multiply $3$ by $2$ .

$\frac{0.25 \cdot 6 h}{h}$

Step 5.1.13

Multiply $0.25$ by $6$ .

$\frac{1.5 h}{h}$

Step 5.2

Cancel the common factor of $h$ .

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Step 5.2.1

Cancel the common factor.

$\frac{1.5 h}{h}$

Step 5.2.2

Divide $1.5$ by $1$ .

$1.5$

Step 6