Precalculus Examples

$f (x) = 2 x + 1$ , $g (x) = 2 x^{2} - 3$ , $(f \cdot g) (2)$

Step 1

Evaluate $f \cdot g$ .

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Step 1.1

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

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

Step 1.2

Expand $(2 x + 1) (2 x^{2} - 3)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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

Rewrite using the commutative property of multiplication.

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

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x^{2}$ .

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

Step 1.3.1.2.2

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

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Step 1.3.1.2.2.1

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

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

Step 1.3.1.2.2.2

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

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

Step 1.3.1.2.3

Add $2$ and $1$ .

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

Step 1.3.1.3

Multiply $2$ by $2$ .

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

Step 1.3.1.4

Multiply $- 3$ by $2$ .

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

Step 1.3.1.5

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

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

Step 1.3.1.6

Multiply $- 3$ by $1$ .

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

Step 1.3.2

Move $- 6 x$ .

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

Step 2

Evaluate $f \cdot g$ at $2$ .

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

Step 3

Simplify $4 {(2)}^{3} + 2 {(2)}^{2} - 6 \cdot 2 - 3$ .

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Step 3.1

Remove parentheses.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Raise $2$ to the power of $3$ .

$4 \cdot 8 + 2 {(2)}^{2} - 6 \cdot 2 - 3$

Step 3.2.2

Multiply $4$ by $8$ .

$32 + 2 {(2)}^{2} - 6 \cdot 2 - 3$

Step 3.2.3

Multiply $2$ by ${(2)}^{2}$ by adding the exponents.

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Step 3.2.3.1

Multiply $2$ by ${(2)}^{2}$ .

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Step 3.2.3.1.1

Raise $2$ to the power of $1$ .

$32 + 2^{1} {(2)}^{2} - 6 \cdot 2 - 3$

Step 3.2.3.1.2

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

$32 + 2^{1 + 2} - 6 \cdot 2 - 3$

Step 3.2.3.2

Add $1$ and $2$ .

$32 + 2^{3} - 6 \cdot 2 - 3$

Step 3.2.4

Raise $2$ to the power of $3$ .

$32 + 8 - 6 \cdot 2 - 3$

Step 3.2.5

Multiply $- 6$ by $2$ .

$32 + 8 - 12 - 3$

Step 3.3

Simplify by adding and subtracting.

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Step 3.3.1

Add $32$ and $8$ .

$40 - 12 - 3$

Step 3.3.2

Subtract $12$ from $40$ .

$28 - 3$

Step 3.3.3

Subtract $3$ from $28$ .

$25$