Precalculus Examples

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

Step 1

Evaluate $f g h$ .

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Step 1.1

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

$(2 x + 2) (2 - x^{2}) (\frac{x}{2} - \frac{6}{2})$

Step 1.2

Expand $(2 x + 2) (2 - x^{2})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$(2 x (2 - x^{2}) + 2 (2 - x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.2.2

Apply the distributive property.

$(2 x \cdot 2 + 2 x (- x^{2}) + 2 (2 - x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.2.3

Apply the distributive property.

$(2 x \cdot 2 + 2 x (- x^{2}) + 2 \cdot 2 + 2 (- x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.3

Simplify terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $2$ by $2$ .

$(4 x + 2 x (- x^{2}) + 2 \cdot 2 + 2 (- x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

$(4 x + 2 \cdot - 1 x \cdot x^{2} + 2 \cdot 2 + 2 (- x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.3.1.3

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.3.1.3.1

Move $x^{2}$ .

$(4 x + 2 \cdot - 1 (x^{2} x) + 2 \cdot 2 + 2 (- x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.3.1.3.2

Multiply $x^{2}$ by $x$ .

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Step 1.3.1.3.2.1

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

$(4 x + 2 \cdot - 1 (x^{2} x^{1}) + 2 \cdot 2 + 2 (- x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.3.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$(4 x + 2 \cdot - 1 x^{2 + 1} + 2 \cdot 2 + 2 (- x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.3.1.3.3

Add $2$ and $1$ .

$(4 x + 2 \cdot - 1 x^{3} + 2 \cdot 2 + 2 (- x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.3.1.4

Multiply $2$ by $- 1$ .

$(4 x - 2 x^{3} + 2 \cdot 2 + 2 (- x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.3.1.5

Multiply $2$ by $2$ .

$(4 x - 2 x^{3} + 4 + 2 (- x^{2})) (\frac{x}{2} - \frac{6}{2})$

Step 1.3.1.6

Multiply $- 1$ by $2$ .

$(4 x - 2 x^{3} + 4 - 2 x^{2}) (\frac{x}{2} - \frac{6}{2})$

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

Divide $6$ by $2$ .

$(4 x - 2 x^{3} + 4 - 2 x^{2}) (\frac{x}{2} - 1 \cdot 3)$

Step 1.3.2.2

Multiply $- 1$ by $3$ .

$(4 x - 2 x^{3} + 4 - 2 x^{2}) (\frac{x}{2} - 3)$

Step 1.4

Expand $(4 x - 2 x^{3} + 4 - 2 x^{2}) (\frac{x}{2} - 3)$ by multiplying each term in the first expression by each term in the second expression.

$4 x \frac{x}{2} + 4 x \cdot - 3 - 2 x^{3} \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5

Simplify terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Cancel the common factor of $2$ .

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Step 1.5.1.1.1

Factor $2$ out of $4 x$ .

$2 (2 x) \frac{x}{2} + 4 x \cdot - 3 - 2 x^{3} \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.1.2

Cancel the common factor.

$2 (2 x) \frac{x}{2} + 4 x \cdot - 3 - 2 x^{3} \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.1.3

Rewrite the expression.

$2 x \cdot x + 4 x \cdot - 3 - 2 x^{3} \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.2

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

$2 (x^{1} x) + 4 x \cdot - 3 - 2 x^{3} \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.3

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

$2 (x^{1} x^{1}) + 4 x \cdot - 3 - 2 x^{3} \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{1 + 1} + 4 x \cdot - 3 - 2 x^{3} \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.5

Add $1$ and $1$ .

$2 x^{2} + 4 x \cdot - 3 - 2 x^{3} \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.6

Multiply $- 3$ by $4$ .

$2 x^{2} - 12 x - 2 x^{3} \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.7

Cancel the common factor of $2$ .

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Step 1.5.1.7.1

Factor $2$ out of $- 2 x^{3}$ .

$2 x^{2} - 12 x + 2 (- x^{3}) \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.7.2

Cancel the common factor.

$2 x^{2} - 12 x + 2 (- x^{3}) \frac{x}{2} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.7.3

Rewrite the expression.

$2 x^{2} - 12 x - x^{3} x - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.8

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

$2 x^{2} - 12 x - (x^{1} x^{3}) - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{2} - 12 x - x^{1 + 3} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.10

Add $1$ and $3$ .

$2 x^{2} - 12 x - x^{4} - 2 x^{3} \cdot - 3 + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.11

Multiply $- 3$ by $- 2$ .

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 4 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.12

Cancel the common factor of $2$ .

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Step 1.5.1.12.1

Factor $2$ out of $4$ .

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 (2) \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.12.2

Cancel the common factor.

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 \cdot 2 \frac{x}{2} + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.12.3

Rewrite the expression.

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 x + 4 \cdot - 3 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.13

Multiply $4$ by $- 3$ .

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 x - 12 - 2 x^{2} \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.14

Cancel the common factor of $2$ .

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Step 1.5.1.14.1

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

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 x - 12 + 2 (- x^{2}) \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.14.2

Cancel the common factor.

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 x - 12 + 2 (- x^{2}) \frac{x}{2} - 2 x^{2} \cdot - 3$

Step 1.5.1.14.3

Rewrite the expression.

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 x - 12 - x^{2} x - 2 x^{2} \cdot - 3$

Step 1.5.1.15

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

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 x - 12 - (x^{1} x^{2}) - 2 x^{2} \cdot - 3$

Step 1.5.1.16

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 x - 12 - x^{1 + 2} - 2 x^{2} \cdot - 3$

Step 1.5.1.17

Add $1$ and $2$ .

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 x - 12 - x^{3} - 2 x^{2} \cdot - 3$

Step 1.5.1.18

Multiply $- 3$ by $- 2$ .

$2 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 x - 12 - x^{3} + 6 x^{2}$

Step 1.5.2

Simplify by adding terms.

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Step 1.5.2.1

Add $2 x^{2}$ and $6 x^{2}$ .

$8 x^{2} - 12 x - x^{4} + 6 x^{3} + 2 x - 12 - x^{3}$

Step 1.5.2.2

Add $- 12 x$ and $2 x$ .

$8 x^{2} - x^{4} + 6 x^{3} - 10 x - 12 - x^{3}$

Step 1.5.2.3

Subtract $x^{3}$ from $6 x^{3}$ .

$8 x^{2} - x^{4} + 5 x^{3} - 10 x - 12$

Step 1.5.2.4

Move $8 x^{2}$ .

$- x^{4} + 5 x^{3} + 8 x^{2} - 10 x - 12$

Step 2

Evaluate $f g h$ at $4$ .

$- {(4)}^{4} + 5 {(4)}^{3} + 8 {(4)}^{2} - 10 \cdot 4 - 12$

Step 3

Simplify $- {(4)}^{4} + 5 {(4)}^{3} + 8 {(4)}^{2} - 10 \cdot 4 - 12$ .

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Step 3.1

Remove parentheses.

$- {(4)}^{4} + 5 {(4)}^{3} + 8 {(4)}^{2} - 10 \cdot 4 - 12$

Step 3.2

Simplify each term.

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Step 3.2.1

Raise $4$ to the power of $4$ .

$- 1 \cdot 256 + 5 {(4)}^{3} + 8 {(4)}^{2} - 10 \cdot 4 - 12$

Step 3.2.2

Multiply $- 1$ by $256$ .

$- 256 + 5 {(4)}^{3} + 8 {(4)}^{2} - 10 \cdot 4 - 12$

Step 3.2.3

Raise $4$ to the power of $3$ .

$- 256 + 5 \cdot 64 + 8 {(4)}^{2} - 10 \cdot 4 - 12$

Step 3.2.4

Multiply $5$ by $64$ .

$- 256 + 320 + 8 {(4)}^{2} - 10 \cdot 4 - 12$

Step 3.2.5

Raise $4$ to the power of $2$ .

$- 256 + 320 + 8 \cdot 16 - 10 \cdot 4 - 12$

Step 3.2.6

Multiply $8$ by $16$ .

$- 256 + 320 + 128 - 10 \cdot 4 - 12$

Step 3.2.7

Multiply $- 10$ by $4$ .

$- 256 + 320 + 128 - 40 - 12$

Step 3.3

Simplify by adding and subtracting.

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Step 3.3.1

Add $- 256$ and $320$ .

$64 + 128 - 40 - 12$

Step 3.3.2

Add $64$ and $128$ .

$192 - 40 - 12$

Step 3.3.3

Subtract $40$ from $192$ .

$152 - 12$

Step 3.3.4

Subtract $12$ from $152$ .

$140$