Algebra Examples

$f (x) = 2 x - 1$

Step 1

Consider the difference quotient formula.

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

Step 2

Find the components of the definition.

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

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

$f (x + h) = 2 (x + h) - 1$

Simplify the result.

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Apply the distributive property.

$f (x + h) = 2 x + 2 h - 1$

The final answer is $2 x + 2 h - 1$ .

$2 x + 2 h - 1$

Reorder $2 x$ and $2 h$ .

$2 h + 2 x - 1$

Find the components of the definition.

$f (x + h) = 2 h + 2 x - 1$

$f (x) = 2 x - 1$

$f (x + h) = 2 h + 2 x - 1$

$f (x) = 2 x - 1$

Step 3

Plug in the components.

$\frac{f (x + h) - f (x)}{h} = \frac{2 h + 2 x - 1 - (2 x - 1)}{h}$

Step 4

Simplify.

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Simplify the numerator.

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Apply the distributive property.

$\frac{2 h + 2 x - 1 - (2 x) - - 1}{h}$

Multiply $2$ by $- 1$ .

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

Multiply $- 1$ by $- 1$ .

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

Subtract $2 x$ from $2 x$ .

$\frac{2 h + 0 - 1 + 1}{h}$

Add $2 h$ and $0$ .

$\frac{2 h - 1 + 1}{h}$

Add $- 1$ and $1$ .

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

Add $2 h$ and $0$ .

$\frac{2 h}{h}$

Divide $2$ by $1$ .

$2$

Step 5