Precalculus Examples

$y = \frac{1}{2} x + 3$

Step 1

Write $y = \frac{1}{2} x + 3$ as a function.

$f (x) = \frac{1}{2} x + 3$

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.

$f (x + h) = \frac{1}{2} \cdot (x + h) + 3$

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.

$f (x + h) = \frac{1}{2} x + \frac{1}{2} h + 3$

Step 3.1.2.1.2

Combine $\frac{1}{2}$ and $x$ .

$f (x + h) = \frac{x}{2} + \frac{1}{2} h + 3$

Step 3.1.2.1.3

Combine $\frac{1}{2}$ and $h$ .

$f (x + h) = \frac{x}{2} + \frac{h}{2} + 3$

Step 3.1.2.2

The final answer is $\frac{x}{2} + \frac{h}{2} + 3$ .

$\frac{x}{2} + \frac{h}{2} + 3$

Step 3.2

Reorder $\frac{x}{2}$ and $\frac{h}{2}$ .

$\frac{h}{2} + \frac{x}{2} + 3$

Step 3.3

Find the components of the definition.

$f (x + h) = \frac{h}{2} + \frac{x}{2} + 3$

$f (x) = \frac{x}{2} + 3$

$f (x + h) = \frac{h}{2} + \frac{x}{2} + 3$

$f (x) = \frac{x}{2} + 3$

Step 4

Plug in the components.

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Multiply $- 1$ by $3$ .

$\frac{\frac{h}{2} + \frac{x}{2} + 3 - \frac{x}{2} - 3}{h}$

Step 5.1.3

Subtract $\frac{x}{2}$ from $\frac{x}{2}$ .

$\frac{\frac{h}{2} + 0 + 3 - 3}{h}$

Step 5.1.4

Add $\frac{h}{2}$ and $0$ .

$\frac{\frac{h}{2} + 3 - 3}{h}$

Step 5.1.5

Subtract $3$ from $3$ .

$\frac{\frac{h}{2} + 0}{h}$

Step 5.1.6

Add $\frac{h}{2}$ and $0$ .

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

Step 5.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3

Cancel the common factor of $h$ .

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Step 5.3.1

Cancel the common factor.

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

Step 5.3.2

Rewrite the expression.

$\frac{1}{2}$

Step 6