Find f(g(x)) f(x)=x^3 g(x)=x^2+3x-7

$f (x) = x^{3}$ $g (x) = x^{2} + 3 x - 7$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x^{2} + 3 x - 7)$ by substituting in the value of $g$ into $f$ .

$f (x^{2} + 3 x - 7) = {(x^{2} + 3 x - 7)}^{3}$

Step 3

Use the Multinomial Theorem.

$f (x^{2} + 3 x - 7) = {(x^{2})}^{3} + 3 {(x^{2})}^{2} (3 x) + 3 x^{2} {(3 x)}^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Step 4

Simplify terms.

Simplify each term.

Multiply the exponents in ${(x^{2})}^{3}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (x^{2} + 3 x - 7) = x^{2 \cdot 3} + 3 {(x^{2})}^{2} (3 x) + 3 x^{2} {(3 x)}^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $2$ by $3$ .

$f (x^{2} + 3 x - 7) = x^{6} + 3 {(x^{2})}^{2} (3 x) + 3 x^{2} {(3 x)}^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Rewrite using the commutative property of multiplication.

$f (x^{2} + 3 x - 7) = x^{6} + 3 \cdot (3 {(x^{2})}^{2} x) + 3 x^{2} {(3 x)}^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $3$ by $3$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 {(x^{2})}^{2} x + 3 x^{2} {(3 x)}^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{2 \cdot 2} x + 3 x^{2} {(3 x)}^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $2$ by $2$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{4} x + 3 x^{2} {(3 x)}^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 (x \cdot x^{4}) + 3 x^{2} {(3 x)}^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $x$ by $x^{4}$ .

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

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

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{1 + 4} + 3 x^{2} {(3 x)}^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Add $1$ and $4$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 3 x^{2} {(3 x)}^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Apply the product rule to $3 x$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 3 x^{2} (3^{2} x^{2}) + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Rewrite using the commutative property of multiplication.

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 3 \cdot (3^{2} x^{2} x^{2}) + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

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

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

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

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 3^{1 + 2} x^{2} x^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Add $1$ and $2$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 3^{3} x^{2} x^{2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

Move $x^{2}$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 3^{3} (x^{2} x^{2}) + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 3^{3} x^{2 + 2} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Add $2$ and $2$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 3^{3} x^{4} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + {(3 x)}^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Apply the product rule to $3 x$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 3^{3} x^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} + 3 {(x^{2})}^{2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply the exponents in ${(x^{2})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} + 3 x^{2 \cdot 2} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $2$ by $2$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} + 3 x^{4} \cdot - 7 + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $- 7$ by $3$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} + 6 x^{2} (3 x) \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

Move $x$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} + 6 (x \cdot x^{2}) \cdot 3 \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

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

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} + 6 (x \cdot x^{2}) \cdot 3 \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} + 6 x^{1 + 2} \cdot 3 \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Add $1$ and $2$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} + 6 x^{3} \cdot 3 \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $3$ by $6$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} + 18 x^{3} \cdot - 7 + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $- 7$ by $18$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} + 3 {(3 x)}^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Apply the product rule to $3 x$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} + 3 (3^{2} x^{2}) \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

Move $3^{2}$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} + 3^{2} \cdot (3 x^{2}) \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

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

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} + 3^{2} \cdot (3 x^{2}) \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} + 3^{2 + 1} x^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Add $2$ and $1$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} + 3^{3} x^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

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

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} + 27 x^{2} \cdot - 7 + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $- 7$ by $27$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} - 189 x^{2} + 3 x^{2} {(- 7)}^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Raise $- 7$ to the power of $2$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} - 189 x^{2} + 3 x^{2} \cdot 49 + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $49$ by $3$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} - 189 x^{2} + 147 x^{2} + 3 (3 x) {(- 7)}^{2} + {(- 7)}^{3}$

Multiply $3$ by $3$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} - 189 x^{2} + 147 x^{2} + 9 x {(- 7)}^{2} + {(- 7)}^{3}$

Raise $- 7$ to the power of $2$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} - 189 x^{2} + 147 x^{2} + 9 x \cdot 49 + {(- 7)}^{3}$

Multiply $49$ by $9$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} - 189 x^{2} + 147 x^{2} + 441 x + {(- 7)}^{3}$

Raise $- 7$ to the power of $3$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 27 x^{4} + 27 x^{3} - 21 x^{4} - 126 x^{3} - 189 x^{2} + 147 x^{2} + 441 x - 343$

Simplify by adding terms.

Subtract $21 x^{4}$ from $27 x^{4}$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 6 x^{4} + 27 x^{3} - 126 x^{3} - 189 x^{2} + 147 x^{2} + 441 x - 343$

Subtract $126 x^{3}$ from $27 x^{3}$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 6 x^{4} - 99 x^{3} - 189 x^{2} + 147 x^{2} + 441 x - 343$

Add $- 189 x^{2}$ and $147 x^{2}$ .

$f (x^{2} + 3 x - 7) = x^{6} + 9 x^{5} + 6 x^{4} - 99 x^{3} - 42 x^{2} + 441 x - 343$

Find f(g(x)) f(x)=x^3 g(x)=x^2+3x-7

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