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

Simplify the result.

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Step 3.1.2.1

Remove parentheses.

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

Step 3.1.2.2

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

$x + h - 2$

Step 3.2

Reorder $x$ and $h$ .

$h + x - 2$

Step 3.3

Find the components of the definition.

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

$f (x) = x - 2$

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

$f (x) = x - 2$

Step 4

Plug in the components.

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Multiply $- 1$ by $- 2$ .

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

Step 5.1.3

Subtract $x$ from $x$ .

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

Step 5.1.4

Add $h$ and $0$ .

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

Step 5.1.5

Add $- 2$ and $2$ .

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

Step 5.1.6

Add $h$ and $0$ .

$\frac{h}{h}$

Step 5.2

Cancel the common factor of $h$ .

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Step 5.2.1

Cancel the common factor.

$\frac{h}{h}$

Step 5.2.2

Rewrite the expression.

$1$

Step 6