Precalculus Examples

$y = 3 x + 2$

Step 1

Write $y = 3 x + 2$ as a function.

$f (x) = 3 x + 2$

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.

$f (x + h) = 3 (x + h) + 2$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Apply the distributive property.

$f (x + h) = 3 x + 3 h + 2$

Step 3.1.2.2

The final answer is $3 x + 3 h + 2$ .

$3 x + 3 h + 2$

$3 x + 3 h + 2$

$3 x + 3 h + 2$

Step 3.2

Reorder $3 x$ and $3 h$ .

$3 h + 3 x + 2$

Step 3.3

Find the components of the definition.

$f (x + h) = 3 h + 3 x + 2$

$f (x) = 3 x + 2$

$f (x + h) = 3 h + 3 x + 2$

$f (x) = 3 x + 2$

Step 4

Plug in the components.

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

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.

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

Step 5.1.2

Multiply $3$ by $- 1$ .

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

Step 5.1.3

Multiply $- 1$ by $2$ .

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

Step 5.1.4

Subtract $3 x$ from $3 x$ .

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

Step 5.1.5

Add $3 h$ and $0$ .

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

Step 5.1.6

Subtract $2$ from $2$ .

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

Step 5.1.7

Add $3 h$ and $0$ .

$\frac{3 h}{h}$

$\frac{3 h}{h}$

Step 5.2

Cancel the common factor of $h$ .

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Step 5.2.1

Cancel the common factor.

$\frac{3 h}{h}$

Step 5.2.2

Divide $3$ by $1$ .

$3$

$3$

$3$

Step 6

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$