Algebra Examples

$f (x) = x^{3} - 27$ , $g (x) = 3 x - 9$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (3 x - 9)$ by substituting in the value of $g$ into $f$ .

$f (3 x - 9) = {(3 x - 9)}^{3} - 27$

Step 3

Simplify each term.

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Step 3.1

Use the Binomial Theorem.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the product rule to $3 x$ .

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

Step 3.2.2

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

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

Step 3.2.3

Apply the product rule to $3 x$ .

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

Step 3.2.4

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

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Step 3.2.4.1

Move $3^{2}$ .

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

Step 3.2.4.2

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

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Step 3.2.4.2.1

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

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

Step 3.2.4.2.2

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

$f (3 x - 9) = 27 x^{3} + 3^{2 + 1} x^{2} \cdot - 9 + 3 (3 x) {(- 9)}^{2} + {(- 9)}^{3} - 27$

Step 3.2.4.3

Add $2$ and $1$ .

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

Step 3.2.5

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

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

Step 3.2.6

Multiply $- 9$ by $27$ .

$f (3 x - 9) = 27 x^{3} - 243 x^{2} + 3 (3 x) {(- 9)}^{2} + {(- 9)}^{3} - 27$

Step 3.2.7

Multiply $3$ by $3$ .

$f (3 x - 9) = 27 x^{3} - 243 x^{2} + 9 x {(- 9)}^{2} + {(- 9)}^{3} - 27$

Step 3.2.8

Raise $- 9$ to the power of $2$ .

$f (3 x - 9) = 27 x^{3} - 243 x^{2} + 9 x \cdot 81 + {(- 9)}^{3} - 27$

Step 3.2.9

Multiply $81$ by $9$ .

$f (3 x - 9) = 27 x^{3} - 243 x^{2} + 729 x + {(- 9)}^{3} - 27$

Step 3.2.10

Raise $- 9$ to the power of $3$ .

$f (3 x - 9) = 27 x^{3} - 243 x^{2} + 729 x - 729 - 27$

Step 4

Subtract $27$ from $- 729$ .

$f (3 x - 9) = 27 x^{3} - 243 x^{2} + 729 x - 756$