Solve the Function Operation f(x)=3x+5 , g(x)=x^3 , gof

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

Set up the composite result function.

$g (f (x))$

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

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

Use the Binomial Theorem.

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

Simplify each term.

Apply the product rule to $3 x$ .

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

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

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

Apply the product rule to $3 x$ .

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

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

Move $3^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

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

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

Multiply $5$ by $27$ .

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

Multiply $3$ by $3$ .

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

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

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

Multiply $25$ by $9$ .

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

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

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