Precalculus Examples

$- 1 + \frac{3}{4} x = y$

Step 1

Write $y = - 1 + \frac{3 x}{4}$ as a function.

$f (x) = - 1 + \frac{3 x}{4}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $y = - 1 + \frac{3}{4} x$ .

$y = - 1 + \frac{3}{4} x$

Step 2.2

Combine $\frac{3}{4}$ and $x$ .

$y = - 1 + \frac{3 x}{4}$

Step 3

Consider the difference quotient formula.

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

Step 4

Find the components of the definition.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Step 4.1.2.2

Simplify terms.

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Step 4.1.2.2.1

Combine $- 1$ and $\frac{4}{4}$ .

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

Step 4.1.2.2.2

Combine the numerators over the common denominator.

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

Step 4.1.2.3

Simplify the numerator.

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Step 4.1.2.3.1

Multiply $- 1$ by $4$ .

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

Step 4.1.2.3.2

Apply the distributive property.

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

Step 4.1.2.4

Simplify with factoring out.

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Step 4.1.2.4.1

Rewrite $- 4$ as $- 1 (4)$ .

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

Step 4.1.2.4.2

Factor $- 1$ out of $3 x$ .

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

Step 4.1.2.4.3

Factor $- 1$ out of $- 1 (4) - (- 3 x)$ .

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

Step 4.1.2.4.4

Factor $- 1$ out of $3 h$ .

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

Step 4.1.2.4.5

Factor $- 1$ out of $- 1 (4 - 3 x) - (- 3 h)$ .

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

Step 4.1.2.4.6

Move the negative in front of the fraction.

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

Step 4.1.2.5

The final answer is $- \frac{4 - 3 x - 3 h}{4}$ .

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

Step 4.2

Find the components of the definition.

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

$f (x) = - 1 + \frac{3 x}{4}$

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

$f (x) = - 1 + \frac{3 x}{4}$

Step 5

Plug in the components.

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Multiply $- 1$ by $- 1$ .

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

Step 6.1.3

Write $1$ as a fraction with a common denominator.

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

Step 6.1.4

Combine the numerators over the common denominator.

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

Step 6.1.5

Combine the numerators over the common denominator.

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

Step 6.1.6

Simplify each term.

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Step 6.1.6.1

Apply the distributive property.

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

Step 6.1.6.2

Simplify.

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Step 6.1.6.2.1

Multiply $- 1$ by $4$ .

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

Step 6.1.6.2.2

Multiply $- 3$ by $- 1$ .

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

Step 6.1.6.2.3

Multiply $- 3$ by $- 1$ .

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

Step 6.1.7

Combine the opposite terms in $- 4 + 3 x + 3 h + 4 - 3 x$ .

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Step 6.1.7.1

Add $- 4$ and $4$ .

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

Step 6.1.7.2

Add $3 x + 3 h$ and $0$ .

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

Step 6.1.7.3

Subtract $3 x$ from $3 x$ .

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

Step 6.1.7.4

Add $3 h$ and $0$ .

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

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{3 h}{4} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Factor $h$ out of $3 h$ .

$\frac{h \cdot 3}{4} \cdot \frac{1}{h}$

Step 6.3.2

Cancel the common factor.

$\frac{h \cdot 3}{4} \cdot \frac{1}{h}$

Step 6.3.3

Rewrite the expression.

$\frac{3}{4}$

Step 7