Algebra Examples

$f (x) = 5 x^{3} - 2$ & $g (x) = 2 x + 1$ ,

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (2 x + 1)$ by substituting in the value of $g$ into $f$ .

$f (2 x + 1) = 5 {(2 x + 1)}^{3} - 2$

Step 3

Simplify each term.

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Step 3.1

Use the Binomial Theorem.

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

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the product rule to $2 x$ .

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

Step 3.2.2

Raise $2$ to the power of $3$ .

$f (2 x + 1) = 5 (8 x^{3} + 3 {(2 x)}^{2} \cdot 1 + 3 (2 x) \cdot 1^{2} + 1^{3}) - 2$

Step 3.2.3

Apply the product rule to $2 x$ .

$f (2 x + 1) = 5 (8 x^{3} + 3 (2^{2} x^{2}) \cdot 1 + 3 (2 x) \cdot 1^{2} + 1^{3}) - 2$

Step 3.2.4

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

$f (2 x + 1) = 5 (8 x^{3} + 3 (4 x^{2}) \cdot 1 + 3 (2 x) \cdot 1^{2} + 1^{3}) - 2$

Step 3.2.5

Multiply $4$ by $3$ .

$f (2 x + 1) = 5 (8 x^{3} + 12 x^{2} \cdot 1 + 3 (2 x) \cdot 1^{2} + 1^{3}) - 2$

Step 3.2.6

Multiply $12$ by $1$ .

$f (2 x + 1) = 5 (8 x^{3} + 12 x^{2} + 3 (2 x) \cdot 1^{2} + 1^{3}) - 2$

Step 3.2.7

Multiply $2$ by $3$ .

$f (2 x + 1) = 5 (8 x^{3} + 12 x^{2} + 6 x \cdot 1^{2} + 1^{3}) - 2$

Step 3.2.8

One to any power is one.

$f (2 x + 1) = 5 (8 x^{3} + 12 x^{2} + 6 x \cdot 1 + 1^{3}) - 2$

Step 3.2.9

Multiply $6$ by $1$ .

$f (2 x + 1) = 5 (8 x^{3} + 12 x^{2} + 6 x + 1^{3}) - 2$

Step 3.2.10

One to any power is one.

$f (2 x + 1) = 5 (8 x^{3} + 12 x^{2} + 6 x + 1) - 2$

Step 3.3

Apply the distributive property.

$f (2 x + 1) = 5 (8 x^{3}) + 5 (12 x^{2}) + 5 (6 x) + 5 \cdot 1 - 2$

Step 3.4

Simplify.

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Step 3.4.1

Multiply $8$ by $5$ .

$f (2 x + 1) = 40 x^{3} + 5 (12 x^{2}) + 5 (6 x) + 5 \cdot 1 - 2$

Step 3.4.2

Multiply $12$ by $5$ .

$f (2 x + 1) = 40 x^{3} + 60 x^{2} + 5 (6 x) + 5 \cdot 1 - 2$

Step 3.4.3

Multiply $6$ by $5$ .

$f (2 x + 1) = 40 x^{3} + 60 x^{2} + 30 x + 5 \cdot 1 - 2$

Step 3.4.4

Multiply $5$ by $1$ .

$f (2 x + 1) = 40 x^{3} + 60 x^{2} + 30 x + 5 - 2$

Step 4

Subtract $2$ from $5$ .

$f (2 x + 1) = 40 x^{3} + 60 x^{2} + 30 x + 3$