Trigonometry Examples

$f (x) = 2 x^{2}$ , $g (x) = 8 x^{2} + 3 x$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (8 x^{2} + 3 x)$ by substituting in the value of $g$ into $f$ .

$f (8 x^{2} + 3 x) = 2 {(8 x^{2} + 3 x)}^{2}$

Step 3

Rewrite ${(8 x^{2} + 3 x)}^{2}$ as $(8 x^{2} + 3 x) (8 x^{2} + 3 x)$ .

$f (8 x^{2} + 3 x) = 2 ((8 x^{2} + 3 x) (8 x^{2} + 3 x))$

Step 4

Expand $(8 x^{2} + 3 x) (8 x^{2} + 3 x)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$f (8 x^{2} + 3 x) = 2 (8 x^{2} (8 x^{2} + 3 x) + 3 x (8 x^{2} + 3 x))$

Step 4.2

Apply the distributive property.

$f (8 x^{2} + 3 x) = 2 (8 x^{2} (8 x^{2}) + 8 x^{2} (3 x) + 3 x (8 x^{2} + 3 x))$

Step 4.3

Apply the distributive property.

$f (8 x^{2} + 3 x) = 2 (8 x^{2} (8 x^{2}) + 8 x^{2} (3 x) + 3 x (8 x^{2}) + 3 x (3 x))$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.1.2

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

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Step 5.1.2.1

Move $x^{2}$ .

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

Step 5.1.2.2

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

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

Step 5.1.2.3

Add $2$ and $2$ .

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

Step 5.1.3

Multiply $8$ by $8$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 8 x^{2} (3 x) + 3 x (8 x^{2}) + 3 x (3 x))$

Step 5.1.4

Rewrite using the commutative property of multiplication.

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 8 \cdot (3 x^{2} x) + 3 x (8 x^{2}) + 3 x (3 x))$

Step 5.1.5

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

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Step 5.1.5.1

Move $x$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 8 \cdot (3 (x \cdot x^{2})) + 3 x (8 x^{2}) + 3 x (3 x))$

Step 5.1.5.2

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

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Step 5.1.5.2.1

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

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 8 \cdot (3 (x \cdot x^{2})) + 3 x (8 x^{2}) + 3 x (3 x))$

Step 5.1.5.2.2

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

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 8 \cdot (3 x^{1 + 2}) + 3 x (8 x^{2}) + 3 x (3 x))$

Step 5.1.5.3

Add $1$ and $2$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 8 \cdot (3 x^{3}) + 3 x (8 x^{2}) + 3 x (3 x))$

Step 5.1.6

Multiply $8$ by $3$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 3 x (8 x^{2}) + 3 x (3 x))$

Step 5.1.7

Rewrite using the commutative property of multiplication.

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 3 \cdot (8 x \cdot x^{2}) + 3 x (3 x))$

Step 5.1.8

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

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Step 5.1.8.1

Move $x^{2}$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 3 \cdot (8 (x^{2} x)) + 3 x (3 x))$

Step 5.1.8.2

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

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Step 5.1.8.2.1

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

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 3 \cdot (8 (x^{2} x)) + 3 x (3 x))$

Step 5.1.8.2.2

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

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 3 \cdot (8 x^{2 + 1}) + 3 x (3 x))$

Step 5.1.8.3

Add $2$ and $1$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 3 \cdot (8 x^{3}) + 3 x (3 x))$

Step 5.1.9

Multiply $3$ by $8$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 24 x^{3} + 3 x (3 x))$

Step 5.1.10

Rewrite using the commutative property of multiplication.

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 24 x^{3} + 3 \cdot (3 x \cdot x))$

Step 5.1.11

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.11.1

Move $x$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 24 x^{3} + 3 \cdot (3 (x \cdot x)))$

Step 5.1.11.2

Multiply $x$ by $x$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 24 x^{3} + 3 \cdot (3 x^{2}))$

Step 5.1.12

Multiply $3$ by $3$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 24 x^{3} + 24 x^{3} + 9 x^{2})$

Step 5.2

Add $24 x^{3}$ and $24 x^{3}$ .

$f (8 x^{2} + 3 x) = 2 (64 x^{4} + 48 x^{3} + 9 x^{2})$

Step 6

Apply the distributive property.

$f (8 x^{2} + 3 x) = 2 (64 x^{4}) + 2 (48 x^{3}) + 2 (9 x^{2})$

Step 7

Simplify.

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Step 7.1

Multiply $64$ by $2$ .

$f (8 x^{2} + 3 x) = 128 x^{4} + 2 (48 x^{3}) + 2 (9 x^{2})$

Step 7.2

Multiply $48$ by $2$ .

$f (8 x^{2} + 3 x) = 128 x^{4} + 96 x^{3} + 2 (9 x^{2})$

Step 7.3

Multiply $9$ by $2$ .

$f (8 x^{2} + 3 x) = 128 x^{4} + 96 x^{3} + 18 x^{2}$