Precalculus Examples

$g (x) = \frac{x^{2}}{4} + 7$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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Step 2.1

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

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Step 2.1.1

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

$g (x + h) = \frac{{(x + h)}^{2}}{4} + 7$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

To write $7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$g (x + h) = \frac{{(x + h)}^{2}}{4} + 7 \cdot \frac{4}{4}$

Step 2.1.2.2

Simplify terms.

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Step 2.1.2.2.1

Combine $7$ and $\frac{4}{4}$ .

$g (x + h) = \frac{{(x + h)}^{2}}{4} + \frac{7 \cdot 4}{4}$

Step 2.1.2.2.2

Combine the numerators over the common denominator.

$g (x + h) = \frac{{(x + h)}^{2} + 7 \cdot 4}{4}$

Step 2.1.2.3

Simplify the numerator.

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Step 2.1.2.3.1

Rewrite ${(x + h)}^{2}$ as $(x + h) (x + h)$ .

$g (x + h) = \frac{(x + h) (x + h) + 7 \cdot 4}{4}$

Step 2.1.2.3.2

Expand $(x + h) (x + h)$ using the FOIL Method.

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Step 2.1.2.3.2.1

Apply the distributive property.

$g (x + h) = \frac{x (x + h) + h (x + h) + 7 \cdot 4}{4}$

Step 2.1.2.3.2.2

Apply the distributive property.

$g (x + h) = \frac{x \cdot x + x h + h (x + h) + 7 \cdot 4}{4}$

Step 2.1.2.3.2.3

Apply the distributive property.

$g (x + h) = \frac{x \cdot x + x h + h x + h \cdot h + 7 \cdot 4}{4}$

Step 2.1.2.3.3

Simplify and combine like terms.

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Step 2.1.2.3.3.1

Simplify each term.

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Step 2.1.2.3.3.1.1

Multiply $x$ by $x$ .

$g (x + h) = \frac{x^{2} + x h + h x + h \cdot h + 7 \cdot 4}{4}$

Step 2.1.2.3.3.1.2

Multiply $h$ by $h$ .

$g (x + h) = \frac{x^{2} + x h + h x + h^{2} + 7 \cdot 4}{4}$

Step 2.1.2.3.3.2

Add $x h$ and $h x$ .

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Step 2.1.2.3.3.2.1

Reorder $x$ and $h$ .

$g (x + h) = \frac{x^{2} + h x + h x + h^{2} + 7 \cdot 4}{4}$

Step 2.1.2.3.3.2.2

Add $h x$ and $h x$ .

$g (x + h) = \frac{x^{2} + 2 h x + h^{2} + 7 \cdot 4}{4}$

Step 2.1.2.3.4

Multiply $7$ by $4$ .

$g (x + h) = \frac{x^{2} + 2 h x + h^{2} + 28}{4}$

Step 2.1.2.4

The final answer is $\frac{x^{2} + 2 h x + h^{2} + 28}{4}$ .

$\frac{x^{2} + 2 h x + h^{2} + 28}{4}$

Step 2.2

Find the components of the definition.

$g (x + h) = \frac{x^{2} + 2 h x + h^{2} + 28}{4}$

$g (x) = \frac{x^{2}}{4} + 7$

$g (x + h) = \frac{x^{2} + 2 h x + h^{2} + 28}{4}$

$g (x) = \frac{x^{2}}{4} + 7$

Step 3

Plug in the components.

$\frac{g (x + h) - g x}{h} = \frac{\frac{x^{2} + 2 h x + h^{2} + 28}{4} - (\frac{x^{2}}{4} + 7)}{h}$

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.

$\frac{\frac{x^{2} + 2 h x + h^{2} + 28}{4} - \frac{x^{2}}{4} - 1 \cdot 7}{h}$

Step 4.1.2

Multiply $- 1$ by $7$ .

$\frac{\frac{x^{2} + 2 h x + h^{2} + 28}{4} - \frac{x^{2}}{4} - 7}{h}$

Step 4.1.3

Combine the numerators over the common denominator.

$\frac{\frac{x^{2} + 2 h x + h^{2} + 28 - x^{2}}{4} - 7}{h}$

Step 4.1.4

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{\frac{x^{2} + 2 h x + h^{2} + 28 - x^{2}}{4} - 7 \cdot \frac{4}{4}}{h}$

Step 4.1.5

Combine $- 7$ and $\frac{4}{4}$ .

$\frac{\frac{x^{2} + 2 h x + h^{2} + 28 - x^{2}}{4} + \frac{- 7 \cdot 4}{4}}{h}$

Step 4.1.6

Combine the numerators over the common denominator.

$\frac{\frac{x^{2} + 2 h x + h^{2} + 28 - x^{2} - 7 \cdot 4}{4}}{h}$

Step 4.1.7

Rewrite $\frac{x^{2} + 2 h x + h^{2} + 28 - x^{2} - 7 \cdot 4}{4}$ in a factored form.

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Step 4.1.7.1

Multiply $- 7$ by $4$ .

$\frac{\frac{x^{2} + 2 h x + h^{2} + 28 - x^{2} - 28}{4}}{h}$

Step 4.1.7.2

Subtract $x^{2}$ from $x^{2}$ .

$\frac{\frac{2 h x + h^{2} + 28 + 0 - 28}{4}}{h}$

Step 4.1.7.3

Add $2 h x$ and $0$ .

$\frac{\frac{2 h x + h^{2} + 28 - 28}{4}}{h}$

Step 4.1.7.4

Subtract $28$ from $28$ .

$\frac{\frac{2 h x + h^{2} + 0}{4}}{h}$

Step 4.1.7.5

Add $2 h x + h^{2}$ and $0$ .

$\frac{\frac{2 h x + h^{2}}{4}}{h}$

Step 4.1.7.6

Factor $h$ out of $2 h x + h^{2}$ .

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Step 4.1.7.6.1

Factor $h$ out of $2 h x$ .

$\frac{\frac{h (2 x) + h^{2}}{4}}{h}$

Step 4.1.7.6.2

Factor $h$ out of $h^{2}$ .

$\frac{\frac{h (2 x) + h (h)}{4}}{h}$

Step 4.1.7.6.3

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

$\frac{\frac{h (2 x + h)}{4}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{h (2 x + h)}{4} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Cancel the common factor.

$\frac{h (2 x + h)}{4} \cdot \frac{1}{h}$

Step 4.3.2

Rewrite the expression.

$\frac{2 x + h}{4}$

Step 5