Precalculus Examples

$f (x) = {(x - 1)}^{2} + 3$ find $\frac{f (x + 2) - f (x - 2)}{x - 1}$

Step 1

Evaluate the numerator in $\frac{f (x + 2) - f (x - 2)}{x - 1}$ .

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Step 1.1

Replace the variable $x$ with $x + 2$ in the expression.

$f (x + 2) = {((x + 2) - 1)}^{2} + 3$

Step 1.2

Replace the variable $x$ with $x - 2$ in the expression.

$f (x - 2) = {((x - 2) - 1)}^{2} + 3$

Step 1.3

Replace the function designators in $f (x + 2) - f (x - 2)$ with the actual functions.

$({((x + 2) - 1)}^{2} + 3) - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4

Simplify $({((x + 2) - 1)}^{2} + 3) - ({((x - 2) - 1)}^{2} + 3)$ .

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Subtract $1$ from $2$ .

${(x + 1)}^{2} + 3 - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4.1.2

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$(x + 1) (x + 1) + 3 - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4.1.3

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.4.1.3.1

Apply the distributive property.

$x (x + 1) + 1 (x + 1) + 3 - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4.1.3.2

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 (x + 1) + 3 - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4.1.3.3

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1 + 3 - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4.1.4

Simplify and combine like terms.

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Step 1.4.1.4.1

Simplify each term.

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Step 1.4.1.4.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 1 + 1 x + 1 \cdot 1 + 3 - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4.1.4.1.2

Multiply $x$ by $1$ .

$x^{2} + x + 1 x + 1 \cdot 1 + 3 - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4.1.4.1.3

Multiply $x$ by $1$ .

$x^{2} + x + x + 1 \cdot 1 + 3 - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4.1.4.1.4

Multiply $1$ by $1$ .

$x^{2} + x + x + 1 + 3 - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4.1.4.2

Add $x$ and $x$ .

$x^{2} + 2 x + 1 + 3 - ({((x - 2) - 1)}^{2} + 3)$

Step 1.4.1.5

Simplify each term.

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Step 1.4.1.5.1

Subtract $1$ from $- 2$ .

$x^{2} + 2 x + 1 + 3 - ({(x - 3)}^{2} + 3)$

Step 1.4.1.5.2

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$x^{2} + 2 x + 1 + 3 - ((x - 3) (x - 3) + 3)$

Step 1.4.1.5.3

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.4.1.5.3.1

Apply the distributive property.

$x^{2} + 2 x + 1 + 3 - (x (x - 3) - 3 (x - 3) + 3)$

Step 1.4.1.5.3.2

Apply the distributive property.

$x^{2} + 2 x + 1 + 3 - (x \cdot x + x \cdot - 3 - 3 (x - 3) + 3)$

Step 1.4.1.5.3.3

Apply the distributive property.

$x^{2} + 2 x + 1 + 3 - (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3 + 3)$

Step 1.4.1.5.4

Simplify and combine like terms.

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Step 1.4.1.5.4.1

Simplify each term.

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Step 1.4.1.5.4.1.1

Multiply $x$ by $x$ .

$x^{2} + 2 x + 1 + 3 - (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3 + 3)$

Step 1.4.1.5.4.1.2

Move $- 3$ to the left of $x$ .

$x^{2} + 2 x + 1 + 3 - (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3 + 3)$

Step 1.4.1.5.4.1.3

Multiply $- 3$ by $- 3$ .

$x^{2} + 2 x + 1 + 3 - (x^{2} - 3 x - 3 x + 9 + 3)$

Step 1.4.1.5.4.2

Subtract $3 x$ from $- 3 x$ .

$x^{2} + 2 x + 1 + 3 - (x^{2} - 6 x + 9 + 3)$

Step 1.4.1.6

Add $9$ and $3$ .

$x^{2} + 2 x + 1 + 3 - (x^{2} - 6 x + 12)$

Step 1.4.1.7

Apply the distributive property.

$x^{2} + 2 x + 1 + 3 - x^{2} - (- 6 x) - 1 \cdot 12$

Step 1.4.1.8

Simplify.

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Step 1.4.1.8.1

Multiply $- 6$ by $- 1$ .

$x^{2} + 2 x + 1 + 3 - x^{2} + 6 x - 1 \cdot 12$

Step 1.4.1.8.2

Multiply $- 1$ by $12$ .

$x^{2} + 2 x + 1 + 3 - x^{2} + 6 x - 12$

Step 1.4.2

Simplify by adding terms.

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Step 1.4.2.1

Combine the opposite terms in $x^{2} + 2 x + 1 + 3 - x^{2} + 6 x - 12$ .

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Step 1.4.2.1.1

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

$2 x + 1 + 3 + 0 + 6 x - 12$

Step 1.4.2.1.2

Add $2 x + 1 + 3$ and $0$ .

$2 x + 1 + 3 + 6 x - 12$

Step 1.4.2.2

Add $2 x$ and $6 x$ .

$8 x + 1 + 3 - 12$

Step 1.4.2.3

Simplify by adding and subtracting.

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Step 1.4.2.3.1

Add $1$ and $3$ .

$8 x + 4 - 12$

Step 1.4.2.3.2

Subtract $12$ from $4$ .

$8 x - 8$

Step 2

Rewrite $\frac{f (x + 2) - f (x - 2)}{x - 1}$ .

$\frac{8 x - 8}{x - 1}$

Step 3

Simplify $\frac{8 x - 8}{x - 1}$ .

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Step 3.1

Factor $8$ out of $8 x - 8$ .

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Step 3.1.1

Factor $8$ out of $8 x$ .

$\frac{8 (x) - 8}{x - 1}$

Step 3.1.2

Factor $8$ out of $- 8$ .

$\frac{8 (x) + 8 (- 1)}{x - 1}$

Step 3.1.3

Factor $8$ out of $8 (x) + 8 (- 1)$ .

$\frac{8 (x - 1)}{x - 1}$

Step 3.2

Cancel the common factor of $x - 1$ .

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Step 3.2.1

Cancel the common factor.

$\frac{8 (x - 1)}{x - 1}$

Step 3.2.2

Divide $8$ by $1$ .

$8$