Precalculus Examples

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

Step 1

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

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

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

Add $\frac{x}{4}$ to both sides of the equation.

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

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.

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

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

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

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

Step 4.1.2.2

Combine the numerators over the common denominator.

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

Step 4.1.2.3

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

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

Step 4.2

Find the components of the definition.

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

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

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

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

Step 5

Plug in the components.

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

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.

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

Step 6.1.2

Multiply $- 1$ by $1$ .

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

Step 6.1.3

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

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

Step 6.1.4

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

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

Step 6.1.5

Combine the numerators over the common denominator.

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

Step 6.1.6

Combine the numerators over the common denominator.

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

Step 6.1.7

Rewrite $\frac{4 + x + h - 1 \cdot 4 - x}{4}$ in a factored form.

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Step 6.1.7.1

Multiply $- 1$ by $4$ .

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

Step 6.1.7.2

Subtract $4$ from $4$ .

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

Step 6.1.7.3

Add $x$ and $0$ .

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

Step 6.1.7.4

Subtract $x$ from $x$ .

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

Step 6.1.7.5

Add $h$ and $0$ .

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

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Cancel the common factor.

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

Step 6.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 7