Precalculus Examples

f (x) = 4 x + 3

$f (x) = 4 x + 3$

Step 1

Consider the difference quotient formula.

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

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

Step 2

Find the components of the definition.

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Step 2.1

Evaluate the function at

x = x + h

$x = x + h$ .

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Step 2.1.1

Replace the variable

x

$x$ with

x + h

$x + h$ in the expression.

f (x + h) = 4 (x + h) + 3

$f (x + h) = 4 (x + h) + 3$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

Apply the distributive property.

f (x + h) = 4 x + 4 h + 3

$f (x + h) = 4 x + 4 h + 3$

Step 2.1.2.2

The final answer is

4 x + 4 h + 3

$4 x + 4 h + 3$ .

4 x + 4 h + 3

$4 x + 4 h + 3$

4 x + 4 h + 3

$4 x + 4 h + 3$

4 x + 4 h + 3

$4 x + 4 h + 3$

Step 2.2

Reorder

4 x

$4 x$ and

4 h

$4 h$ .

4 h + 4 x + 3

$4 h + 4 x + 3$

Step 2.3

Find the components of the definition.

f (x + h) = 4 h + 4 x + 3

$f (x + h) = 4 h + 4 x + 3$

f (x) = 4 x + 3

$f (x) = 4 x + 3$

f (x + h) = 4 h + 4 x + 3

$f (x + h) = 4 h + 4 x + 3$

f (x) = 4 x + 3

$f (x) = 4 x + 3$

Step 3

Plug in the components.

\frac{f (x + h) - f (x)}{h} = \frac{4 h + 4 x + 3 - (4 x + 3)}{h}

$\frac{f (x + h) - f (x)}{h} = \frac{4 h + 4 x + 3 - (4 x + 3)}{h}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the distributive property.

\frac{4 h + 4 x + 3 - (4 x) - 1 \cdot 3}{h}

$\frac{4 h + 4 x + 3 - (4 x) - 1 \cdot 3}{h}$

Step 4.1.2

Multiply

4

$4$ by

- 1

$- 1$ .

\frac{4 h + 4 x + 3 - 4 x - 1 \cdot 3}{h}

$\frac{4 h + 4 x + 3 - 4 x - 1 \cdot 3}{h}$

Step 4.1.3

Multiply

- 1

$- 1$ by

3

$3$ .

\frac{4 h + 4 x + 3 - 4 x - 3}{h}

$\frac{4 h + 4 x + 3 - 4 x - 3}{h}$

Step 4.1.4

Subtract

4 x

$4 x$ from

4 x

$4 x$ .

\frac{4 h + 0 + 3 - 3}{h}

$\frac{4 h + 0 + 3 - 3}{h}$

Step 4.1.5

Add

4 h

$4 h$ and

0

$0$ .

\frac{4 h + 3 - 3}{h}

$\frac{4 h + 3 - 3}{h}$

Step 4.1.6

Subtract

3

$3$ from

3

$3$ .

\frac{4 h + 0}{h}

$\frac{4 h + 0}{h}$

Step 4.1.7

Add

4 h

$4 h$ and

0

$0$ .

\frac{4 h}{h}

$\frac{4 h}{h}$

\frac{4 h}{h}

$\frac{4 h}{h}$

Step 4.2

Cancel the common factor of

h

$h$ .

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Step 4.2.1

Cancel the common factor.

\frac{4 h}{h}

$\frac{4 h}{h}$

Step 4.2.2

Divide

4

$4$ by

1

$1$ .

4

$4$

4

$4$

4

$4$

Step 5

f (x) = 4 x + 3

$f (x) = 4 x + 3$

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

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