Precalculus Examples

$y = | x | + 2$

Step 1

Write $y = | x | + 2$ as a function.

$f (x) = | x | + 2$

Step 2

Consider the difference quotient formula.

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

Step 3

Find the components of the definition.

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Step 3.1

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

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Step 3.1.1

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

$f (x + h) = | x + h | + 2$

Step 3.1.2

The final answer is $| x + h | + 2$ .

$| x + h | + 2$

Step 3.2

Find the components of the definition.

$f (x + h) = | x + h | + 2$

$f (x) = | x | + 2$

$f (x + h) = | x + h | + 2$

$f (x) = | x | + 2$

Step 4

Plug in the components.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 1$ by $2$ .

$\frac{| x + h | + 2 - | x | - 2}{h}$

Step 5.3

Subtract $2$ from $2$ .

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

Step 5.4

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

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

Step 6