Precalculus Examples

$f (n) = n^{2} + 3 n$ $g (n) = - n - 3$ Find $f (g (3))$

Step 1

Set up the composite result function.

$f (g (n))$

Step 2

Evaluate $f (- n - 3)$ by substituting in the value of $g$ into $f$ .

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

Step 3

Simplify each term.

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Step 3.1

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

$f (- n - 3) = (- n - 3) (- n - 3) + 3 (- n - 3)$

Step 3.2

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

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Step 3.2.1

Apply the distributive property.

$f (- n - 3) = - n (- n - 3) - 3 (- n - 3) + 3 (- n - 3)$

Step 3.2.2

Apply the distributive property.

$f (- n - 3) = - n (- n) - n \cdot - 3 - 3 (- n - 3) + 3 (- n - 3)$

Step 3.2.3

Apply the distributive property.

$f (- n - 3) = - n (- n) - n \cdot - 3 - 3 (- n) - 3 \cdot - 3 + 3 (- n - 3)$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$f (- n - 3) = - 1 \cdot (- 1 n \cdot n) - n \cdot - 3 - 3 (- n) - 3 \cdot - 3 + 3 (- n - 3)$

Step 3.3.1.2

Multiply $n$ by $n$ by adding the exponents.

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Step 3.3.1.2.1

Move $n$ .

$f (- n - 3) = - 1 \cdot (- 1 (n \cdot n)) - n \cdot - 3 - 3 (- n) - 3 \cdot - 3 + 3 (- n - 3)$

Step 3.3.1.2.2

Multiply $n$ by $n$ .

$f (- n - 3) = - 1 \cdot (- 1 n^{2}) - n \cdot - 3 - 3 (- n) - 3 \cdot - 3 + 3 (- n - 3)$

Step 3.3.1.3

Multiply $- 1$ by $- 1$ .

$f (- n - 3) = 1 n^{2} - n \cdot - 3 - 3 (- n) - 3 \cdot - 3 + 3 (- n - 3)$

Step 3.3.1.4

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

$f (- n - 3) = n^{2} - n \cdot - 3 - 3 (- n) - 3 \cdot - 3 + 3 (- n - 3)$

Step 3.3.1.5

Multiply $- 3$ by $- 1$ .

$f (- n - 3) = n^{2} + 3 n - 3 (- n) - 3 \cdot - 3 + 3 (- n - 3)$

Step 3.3.1.6

Multiply $- 1$ by $- 3$ .

$f (- n - 3) = n^{2} + 3 n + 3 n - 3 \cdot - 3 + 3 (- n - 3)$

Step 3.3.1.7

Multiply $- 3$ by $- 3$ .

$f (- n - 3) = n^{2} + 3 n + 3 n + 9 + 3 (- n - 3)$

Step 3.3.2

Add $3 n$ and $3 n$ .

$f (- n - 3) = n^{2} + 6 n + 9 + 3 (- n - 3)$

Step 3.4

Apply the distributive property.

$f (- n - 3) = n^{2} + 6 n + 9 + 3 (- n) + 3 \cdot - 3$

Step 3.5

Multiply $- 1$ by $3$ .

$f (- n - 3) = n^{2} + 6 n + 9 - 3 n + 3 \cdot - 3$

Step 3.6

Multiply $3$ by $- 3$ .

$f (- n - 3) = n^{2} + 6 n + 9 - 3 n - 9$

Step 4

Simplify by adding terms.

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Step 4.1

Combine the opposite terms in $n^{2} + 6 n + 9 - 3 n - 9$ .

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Step 4.1.1

Subtract $9$ from $9$ .

$f (- n - 3) = n^{2} + 6 n - 3 n + 0$

Step 4.1.2

Add $n^{2} + 6 n - 3 n$ and $0$ .

$f (- n - 3) = n^{2} + 6 n - 3 n$

Step 4.2

Subtract $3 n$ from $6 n$ .

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

Step 5

Evaluate the result function by replacing $n$ with $3$ in the function.

${(3)}^{2} + 3 (3)$

Step 6

Simplify the evaluated function.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$9 + 3 (3)$

Step 6.1.2

Multiply $3$ by $3$ .

$9 + 9$

Step 6.2

Add $9$ and $9$ .

$18$