Precalculus Examples

$f (x) = x^{2} - x - 4$ , $g (x) = 2 x - 6$ , $\frac{f (x - h) - f (x)}{h}$

Step 1

Evaluate the numerator in $\frac{f (x - h) - f (x)}{h}$ .

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Step 1.1

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

$f (x - h) = {(x - h)}^{2} - (x - h) - 4$

Step 1.2

Replace the function designators in $f (x - h) - f (x)$ with the actual functions.

$({(x - h)}^{2} - (x - h) - 4) - (x^{2} - x - 4)$

Step 1.3

Simplify $({(x - h)}^{2} - (x - h) - 4) - (x^{2} - x - 4)$ .

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite ${(x - h)}^{2}$ as $(x - h) (x - h)$ .

$(x - h) (x - h) - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.2

Expand $(x - h) (x - h)$ using the FOIL Method.

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Step 1.3.1.2.1

Apply the distributive property.

$x (x - h) - h (x - h) - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.2.2

Apply the distributive property.

$x \cdot x + x (- h) - h (x - h) - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.2.3

Apply the distributive property.

$x \cdot x + x (- h) - h x - h (- h) - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.3

Simplify and combine like terms.

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Step 1.3.1.3.1

Simplify each term.

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Step 1.3.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x (- h) - h x - h (- h) - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.3.1.2

Rewrite using the commutative property of multiplication.

$x^{2} - x h - h x - h (- h) - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.3.1.3

Rewrite using the commutative property of multiplication.

$x^{2} - x h - h x - 1 \cdot - 1 h \cdot h - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.3.1.4

Multiply $h$ by $h$ by adding the exponents.

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Step 1.3.1.3.1.4.1

Move $h$ .

$x^{2} - x h - h x - 1 \cdot - 1 (h \cdot h) - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.3.1.4.2

Multiply $h$ by $h$ .

$x^{2} - x h - h x - 1 \cdot - 1 h^{2} - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.3.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} - x h - h x + 1 h^{2} - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.3.1.6

Multiply $h^{2}$ by $1$ .

$x^{2} - x h - h x + h^{2} - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.3.2

Subtract $h x$ from $- x h$ .

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Step 1.3.1.3.2.1

Move $x$ .

$x^{2} - h x - h x + h^{2} - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.3.2.2

Subtract $h x$ from $- h x$ .

$x^{2} - 2 h x + h^{2} - (x - h) - 4 - (x^{2} - x - 4)$

Step 1.3.1.4

Apply the distributive property.

$x^{2} - 2 h x + h^{2} - x - - h - 4 - (x^{2} - x - 4)$

Step 1.3.1.5

Multiply $- - h$ .

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Step 1.3.1.5.1

Multiply $- 1$ by $- 1$ .

$x^{2} - 2 h x + h^{2} - x + 1 h - 4 - (x^{2} - x - 4)$

Step 1.3.1.5.2

Multiply $h$ by $1$ .

$x^{2} - 2 h x + h^{2} - x + h - 4 - (x^{2} - x - 4)$

Step 1.3.1.6

Apply the distributive property.

$x^{2} - 2 h x + h^{2} - x + h - 4 - x^{2} - - x - - 4$

Step 1.3.1.7

Simplify.

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Step 1.3.1.7.1

Multiply $- - x$ .

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Step 1.3.1.7.1.1

Multiply $- 1$ by $- 1$ .

$x^{2} - 2 h x + h^{2} - x + h - 4 - x^{2} + 1 x - - 4$

Step 1.3.1.7.1.2

Multiply $x$ by $1$ .

$x^{2} - 2 h x + h^{2} - x + h - 4 - x^{2} + x - - 4$

Step 1.3.1.7.2

Multiply $- 1$ by $- 4$ .

$x^{2} - 2 h x + h^{2} - x + h - 4 - x^{2} + x + 4$

Step 1.3.2

Combine the opposite terms in $x^{2} - 2 h x + h^{2} - x + h - 4 - x^{2} + x + 4$ .

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Step 1.3.2.1

Subtract $x^{2}$ from $x^{2}$ .

$- 2 h x + h^{2} - x + h - 4 + 0 + x + 4$

Step 1.3.2.2

Add $- 2 h x + h^{2} - x + h - 4$ and $0$ .

$- 2 h x + h^{2} - x + h - 4 + x + 4$

Step 1.3.2.3

Add $- x$ and $x$ .

$- 2 h x + h^{2} + h - 4 + 0 + 4$

Step 1.3.2.4

Add $- 2 h x + h^{2} + h - 4$ and $0$ .

$- 2 h x + h^{2} + h - 4 + 4$

Step 1.3.2.5

Add $- 4$ and $4$ .

$- 2 h x + h^{2} + h + 0$

Step 1.3.2.6

Add $- 2 h x + h^{2} + h$ and $0$ .

$- 2 h x + h^{2} + h$

Step 2

Replace the function designators in $h$ with the actual functions.

$h$

Step 3

Rewrite $\frac{f (x - h) - f (x)}{h}$ .

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

Step 4

Simplify $\frac{- 2 h x + h^{2} + h}{h}$ .

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Step 4.1

Factor $h$ out of $- 2 h x + h^{2} + h$ .

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Step 4.1.1

Factor $h$ out of $- 2 h x$ .

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

Step 4.1.2

Factor $h$ out of $h^{2}$ .

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

Step 4.1.3

Raise $h$ to the power of $1$ .

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

Step 4.1.4

Factor $h$ out of $h^{1}$ .

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

Step 4.1.5

Factor $h$ out of $h (- 2 x) + h \cdot h$ .

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

Step 4.1.6

Factor $h$ out of $h (- 2 x + h) + h \cdot 1$ .

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

Step 4.2

Cancel the common factor of $h$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Divide $- 2 x + h + 1$ by $1$ .

$- 2 x + h + 1$

Step 4.3

Reorder $- 2 x$ and $h$ .

$h - 2 x + 1$