Find g(f(x)) f(x)=-3x-3 , g(x)=-3x^3-2

$f (x) = - 3 x - 3$ , $g (x) = - 3 x^{3} - 2$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

$g (- 3 x - 3) = - 3 {(- 3 x - 3)}^{3} - 2$

Step 3

Simplify each term.

Use the Binomial Theorem.

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

Simplify each term.

Apply the product rule to $- 3 x$ .

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

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

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

Apply the product rule to $- 3 x$ .

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

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

Move $- 3$ .

$g (- 3 x - 3) = - 3 (- 27 x^{3} + 3 (- 3 {(- 3)}^{2} x^{2}) + 3 (- 3 x) {(- 3)}^{2} + {(- 3)}^{3}) - 2$

Multiply $- 3$ by ${(- 3)}^{2}$ .

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

$g (- 3 x - 3) = - 3 (- 27 x^{3} + 3 ((- 3) {(- 3)}^{2} x^{2}) + 3 (- 3 x) {(- 3)}^{2} + {(- 3)}^{3}) - 2$

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

$g (- 3 x - 3) = - 3 (- 27 x^{3} + 3 ({(- 3)}^{1 + 2} x^{2}) + 3 (- 3 x) {(- 3)}^{2} + {(- 3)}^{3}) - 2$

Add $1$ and $2$ .

$g (- 3 x - 3) = - 3 (- 27 x^{3} + 3 ({(- 3)}^{3} x^{2}) + 3 (- 3 x) {(- 3)}^{2} + {(- 3)}^{3}) - 2$

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

$g (- 3 x - 3) = - 3 (- 27 x^{3} + 3 (- 27 x^{2}) + 3 (- 3 x) {(- 3)}^{2} + {(- 3)}^{3}) - 2$

Multiply $- 27$ by $3$ .

$g (- 3 x - 3) = - 3 (- 27 x^{3} - 81 x^{2} + 3 (- 3 x) {(- 3)}^{2} + {(- 3)}^{3}) - 2$

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

Move ${(- 3)}^{2}$ .

$g (- 3 x - 3) = - 3 (- 27 x^{3} - 81 x^{2} + 3 ({(- 3)}^{2} \cdot (- 3 x)) + {(- 3)}^{3}) - 2$

Multiply ${(- 3)}^{2}$ by $- 3$ .

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

$g (- 3 x - 3) = - 3 (- 27 x^{3} - 81 x^{2} + 3 ({(- 3)}^{2} \cdot ((- 3) x)) + {(- 3)}^{3}) - 2$

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

$g (- 3 x - 3) = - 3 (- 27 x^{3} - 81 x^{2} + 3 ({(- 3)}^{2 + 1} x) + {(- 3)}^{3}) - 2$

Add $2$ and $1$ .

$g (- 3 x - 3) = - 3 (- 27 x^{3} - 81 x^{2} + 3 ({(- 3)}^{3} x) + {(- 3)}^{3}) - 2$

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

$g (- 3 x - 3) = - 3 (- 27 x^{3} - 81 x^{2} + 3 (- 27 x) + {(- 3)}^{3}) - 2$

Multiply $- 27$ by $3$ .

$g (- 3 x - 3) = - 3 (- 27 x^{3} - 81 x^{2} - 81 x + {(- 3)}^{3}) - 2$

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

$g (- 3 x - 3) = - 3 (- 27 x^{3} - 81 x^{2} - 81 x - 27) - 2$

Apply the distributive property.

$g (- 3 x - 3) = - 3 (- 27 x^{3}) - 3 (- 81 x^{2}) - 3 (- 81 x) - 3 \cdot - 27 - 2$

Simplify.

Multiply $- 27$ by $- 3$ .

$g (- 3 x - 3) = 81 x^{3} - 3 (- 81 x^{2}) - 3 (- 81 x) - 3 \cdot - 27 - 2$

Multiply $- 81$ by $- 3$ .

$g (- 3 x - 3) = 81 x^{3} + 243 x^{2} - 3 (- 81 x) - 3 \cdot - 27 - 2$

Multiply $- 81$ by $- 3$ .

$g (- 3 x - 3) = 81 x^{3} + 243 x^{2} + 243 x - 3 \cdot - 27 - 2$

Multiply $- 3$ by $- 27$ .

$g (- 3 x - 3) = 81 x^{3} + 243 x^{2} + 243 x + 81 - 2$

Step 4

Subtract $2$ from $81$ .

$g (- 3 x - 3) = 81 x^{3} + 243 x^{2} + 243 x + 79$

Find g(f(x)) f(x)=-3x-3 , g(x)=-3x^3-2

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