Solve the Function Operation

$f (x) = 3 x + 5$ , $g (x) = x^{3}$ , $(g \circ f)$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (3 x + 5)$ by substituting in the value of $f$ into $g$ .

$g (3 x + 5) = {(3 x + 5)}^{3}$

Step 3

Use the Binomial Theorem.

$g (3 x + 5) = {(3 x)}^{3} + 3 {(3 x)}^{2} \cdot 5 + 3 (3 x) \cdot 5^{2} + 5^{3}$

Step 4

Simplify each term.

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Step 4.1

Apply the product rule to $3 x$ .

$g (3 x + 5) = 3^{3} x^{3} + 3 {(3 x)}^{2} \cdot 5 + 3 (3 x) \cdot 5^{2} + 5^{3}$

Step 4.2

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

$g (3 x + 5) = 27 x^{3} + 3 {(3 x)}^{2} \cdot 5 + 3 (3 x) \cdot 5^{2} + 5^{3}$

Step 4.3

Apply the product rule to $3 x$ .

$g (3 x + 5) = 27 x^{3} + 3 (3^{2} x^{2}) \cdot 5 + 3 (3 x) \cdot 5^{2} + 5^{3}$

Step 4.4

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 4.4.1

Move $3^{2}$ .

$g (3 x + 5) = 27 x^{3} + 3^{2} \cdot (3 x^{2}) \cdot 5 + 3 (3 x) \cdot 5^{2} + 5^{3}$

Step 4.4.2

Multiply $3^{2}$ by $3$ .

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Step 4.4.2.1

Raise $3$ to the power of $1$ .

$g (3 x + 5) = 27 x^{3} + 3^{2} \cdot (3 x^{2}) \cdot 5 + 3 (3 x) \cdot 5^{2} + 5^{3}$

Step 4.4.2.2

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

$g (3 x + 5) = 27 x^{3} + 3^{2 + 1} x^{2} \cdot 5 + 3 (3 x) \cdot 5^{2} + 5^{3}$

Step 4.4.3

Add $2$ and $1$ .

$g (3 x + 5) = 27 x^{3} + 3^{3} x^{2} \cdot 5 + 3 (3 x) \cdot 5^{2} + 5^{3}$

Step 4.5

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

$g (3 x + 5) = 27 x^{3} + 27 x^{2} \cdot 5 + 3 (3 x) \cdot 5^{2} + 5^{3}$

Step 4.6

Multiply $5$ by $27$ .

$g (3 x + 5) = 27 x^{3} + 135 x^{2} + 3 (3 x) \cdot 5^{2} + 5^{3}$

Step 4.7

Multiply $3$ by $3$ .

$g (3 x + 5) = 27 x^{3} + 135 x^{2} + 9 x \cdot 5^{2} + 5^{3}$

Step 4.8

Raise $5$ to the power of $2$ .

$g (3 x + 5) = 27 x^{3} + 135 x^{2} + 9 x \cdot 25 + 5^{3}$

Step 4.9

Multiply $25$ by $9$ .

$g (3 x + 5) = 27 x^{3} + 135 x^{2} + 225 x + 5^{3}$

Step 4.10

Raise $5$ to the power of $3$ .

$g (3 x + 5) = 27 x^{3} + 135 x^{2} + 225 x + 125$

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