Precalculus Examples

$3 y = 2 x + 5$

Step 1

Write $y = \frac{2 x}{3} + \frac{5}{3}$ as a function.

$f (x) = \frac{2 x}{3} + \frac{5}{3}$

Step 2

Divide each term in $3 y = 2 x + 5$ by $3$ and simplify.

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Step 2.1

Divide each term in $3 y = 2 x + 5$ by $3$ .

$\frac{3 y}{3} = \frac{2 x}{3} + \frac{5}{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{2 x}{3} + \frac{5}{3}$

Step 2.2.1.2

Divide $y$ by $1$ .

$y = \frac{2 x}{3} + \frac{5}{3}$

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.

$f (x + h) = \frac{2 (x + h)}{3} + \frac{5}{3}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Combine the numerators over the common denominator.

$f (x + h) = \frac{2 (x + h) + 5}{3}$

Step 4.1.2.2

Apply the distributive property.

$f (x + h) = \frac{2 x + 2 h + 5}{3}$

Step 4.1.2.3

The final answer is $\frac{2 x + 2 h + 5}{3}$ .

$\frac{2 x + 2 h + 5}{3}$

Step 4.2

Find the components of the definition.

$f (x + h) = \frac{2 x + 2 h + 5}{3}$

$f (x) = \frac{2 x}{3} + \frac{5}{3}$

$f (x + h) = \frac{2 x + 2 h + 5}{3}$

$f (x) = \frac{2 x}{3} + \frac{5}{3}$

Step 5

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{\frac{2 x + 2 h + 5}{3} - (\frac{2 x}{3} + \frac{5}{3})}{h}$

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.

$\frac{\frac{2 x + 2 h + 5}{3} - \frac{2 x}{3} - \frac{5}{3}}{h}$

Step 6.1.2

Combine the numerators over the common denominator.

$\frac{\frac{2 x + 2 h + 5 - 2 x}{3} - \frac{5}{3}}{h}$

Step 6.1.3

Combine the numerators over the common denominator.

$\frac{\frac{2 x + 2 h + 5 - 2 x - 5}{3}}{h}$

Step 6.1.4

Rewrite $\frac{2 x + 2 h + 5 - 2 x - 5}{3}$ in a factored form.

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Step 6.1.4.1

Subtract $2 x$ from $2 x$ .

$\frac{\frac{2 h + 5 + 0 - 5}{3}}{h}$

Step 6.1.4.2

Add $2 h$ and $0$ .

$\frac{\frac{2 h + 5 - 5}{3}}{h}$

Step 6.1.4.3

Subtract $5$ from $5$ .

$\frac{\frac{2 h + 0}{3}}{h}$

Step 6.1.4.4

Add $2 h$ and $0$ .

$\frac{\frac{2 h}{3}}{h}$

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 h}{3} \cdot \frac{1}{h}$

Step 6.3

Cancel the common factor of $h$ .

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Step 6.3.1

Factor $h$ out of $2 h$ .

$\frac{h \cdot 2}{3} \cdot \frac{1}{h}$

Step 6.3.2

Cancel the common factor.

$\frac{h \cdot 2}{3} \cdot \frac{1}{h}$

Step 6.3.3

Rewrite the expression.

$\frac{2}{3}$

Step 7