Precalculus Examples

$g (x) = \frac{5}{x - 1} + 2$

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.

$g (x + h) = \frac{5}{(x + h) - 1} + 2$

Step 2.1.2

Simplify the result.

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Step 2.1.2.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{x + h - 1}{x + h - 1}$ .

$g (x + h) = \frac{5}{x + h - 1} + \frac{2 (x + h - 1)}{x + h - 1}$

Step 2.1.2.2

Combine the numerators over the common denominator.

$g (x + h) = \frac{5 + 2 (x + h - 1)}{x + h - 1}$

Step 2.1.2.3

Simplify the numerator.

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Step 2.1.2.3.1

Apply the distributive property.

$g (x + h) = \frac{5 + 2 x + 2 h + 2 \cdot - 1}{x + h - 1}$

Step 2.1.2.3.2

Multiply $2$ by $- 1$ .

$g (x + h) = \frac{5 + 2 x + 2 h - 2}{x + h - 1}$

Step 2.1.2.3.3

Subtract $2$ from $5$ .

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

Step 2.1.2.4

The final answer is $\frac{2 x + 2 h + 3}{x + h - 1}$ .

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

Step 2.2

Find the components of the definition.

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

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

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

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

Step 3

Plug in the components.

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

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

To write $\frac{2 x + 2 h + 3}{x + h - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 4.1.2

To write $- \frac{2 x + 3}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + h - 1}{x + h - 1}$ .

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

Step 4.1.3

Write each expression with a common denominator of $(x + h - 1) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.3.1

Multiply $\frac{2 x + 2 h + 3}{x + h - 1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 4.1.3.2

Multiply $\frac{2 x + 3}{x - 1}$ by $\frac{x + h - 1}{x + h - 1}$ .

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

Step 4.1.3.3

Reorder the factors of $(x - 1) (x + h - 1)$ .

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

Step 4.1.4

Combine the numerators over the common denominator.

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

Step 4.1.5

Rewrite $\frac{(2 x + 2 h + 3) (x - 1) - (2 x + 3) (x + h - 1)}{(x + h - 1) (x - 1)}$ in a factored form.

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Step 4.1.5.1

Expand $(2 x + 2 h + 3) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.1.5.2

Simplify each term.

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Step 4.1.5.2.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.5.2.1.1

Move $x$ .

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

Step 4.1.5.2.1.2

Multiply $x$ by $x$ .

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

Step 4.1.5.2.2

Multiply $- 1$ by $2$ .

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

Step 4.1.5.2.3

Multiply $- 1$ by $2$ .

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

Step 4.1.5.2.4

Multiply $3$ by $- 1$ .

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

Step 4.1.5.3

Add $- 2 x$ and $3 x$ .

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

Step 4.1.5.4

Apply the distributive property.

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

Step 4.1.5.5

Multiply $2$ by $- 1$ .

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

Step 4.1.5.6

Multiply $- 1$ by $3$ .

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

Step 4.1.5.7

Expand $(- 2 x - 3) (x + h - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4.1.5.8

Simplify each term.

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Step 4.1.5.8.1

Multiply $x$ by $x$ by adding the exponents.

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Step 4.1.5.8.1.1

Move $x$ .

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

Step 4.1.5.8.1.2

Multiply $x$ by $x$ .

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

Step 4.1.5.8.2

Multiply $- 1$ by $- 2$ .

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

Step 4.1.5.8.3

Multiply $- 3$ by $- 1$ .

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

Step 4.1.5.9

Subtract $3 x$ from $2 x$ .

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

Step 4.1.5.10

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

$\frac{\frac{x + 2 h x - 2 h - 3 + 0 - 2 x h - x - 3 h + 3}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.11

Add $x$ and $0$ .

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

Step 4.1.5.12

Subtract $x$ from $x$ .

$\frac{\frac{2 h x - 2 h - 3 - 2 x h + 0 - 3 h + 3}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.13

Subtract $2 x h$ from $2 h x$ .

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Step 4.1.5.13.1

Move $h$ .

$\frac{\frac{- 2 h - 3 + 2 x h - 2 x h + 0 - 3 h + 3}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.13.2

Subtract $2 x h$ from $2 x h$ .

$\frac{\frac{- 2 h - 3 + 0 + 0 - 3 h + 3}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.14

Add $- 2 h$ and $0$ .

$\frac{\frac{- 2 h - 3 + 0 - 3 h + 3}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.15

Add $- 2 h$ and $0$ .

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

Step 4.1.5.16

Subtract $3 h$ from $- 2 h$ .

$\frac{\frac{- 5 h - 3 + 3}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.17

Add $- 3$ and $3$ .

$\frac{\frac{- 5 h + 0}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.5.18

Add $- 5 h$ and $0$ .

$\frac{\frac{- 5 h}{(x + h - 1) (x - 1)}}{h}$

Step 4.1.6

Move the negative in front of the fraction.

$\frac{- \frac{5 h}{(x + h - 1) (x - 1)}}{h}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$- \frac{5 h}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor of $h$ .

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Step 4.3.1

Move the leading negative in $- \frac{5 h}{(x + h - 1) (x - 1)}$ into the numerator.

$\frac{- 5 h}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3.2

Factor $h$ out of $- 5 h$ .

$\frac{h \cdot - 5}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3.3

Cancel the common factor.

$\frac{h \cdot - 5}{(x + h - 1) (x - 1)} \cdot \frac{1}{h}$

Step 4.3.4

Rewrite the expression.

$\frac{- 5}{(x + h - 1) (x - 1)}$

Step 4.4

Move the negative in front of the fraction.

$- \frac{5}{(x + h - 1) (x - 1)}$

Step 5