Algebra Examples

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

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

Step 3

Simplify each term.

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Step 3.1

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

$g (3 x + 2) = 2 ((3 x + 2) (3 x + 2)) - 1$

Step 3.2

Expand $(3 x + 2) (3 x + 2)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$g (3 x + 2) = 2 (3 x (3 x + 2) + 2 (3 x + 2)) - 1$

Step 3.2.2

Apply the distributive property.

$g (3 x + 2) = 2 (3 x (3 x) + 3 x \cdot 2 + 2 (3 x + 2)) - 1$

Step 3.2.3

Apply the distributive property.

$g (3 x + 2) = 2 (3 x (3 x) + 3 x \cdot 2 + 2 (3 x) + 2 \cdot 2) - 1$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$g (3 x + 2) = 2 (3 \cdot (3 x \cdot x) + 3 x \cdot 2 + 2 (3 x) + 2 \cdot 2) - 1$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

$g (3 x + 2) = 2 (3 \cdot (3 (x \cdot x)) + 3 x \cdot 2 + 2 (3 x) + 2 \cdot 2) - 1$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.3.1.3

Multiply $3$ by $3$ .

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

Step 3.3.1.4

Multiply $2$ by $3$ .

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

Step 3.3.1.5

Multiply $3$ by $2$ .

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

Step 3.3.1.6

Multiply $2$ by $2$ .

$g (3 x + 2) = 2 (9 x^{2} + 6 x + 6 x + 4) - 1$

Step 3.3.2

Add $6 x$ and $6 x$ .

$g (3 x + 2) = 2 (9 x^{2} + 12 x + 4) - 1$

Step 3.4

Apply the distributive property.

$g (3 x + 2) = 2 (9 x^{2}) + 2 (12 x) + 2 \cdot 4 - 1$

Step 3.5

Simplify.

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Step 3.5.1

Multiply $9$ by $2$ .

$g (3 x + 2) = 18 x^{2} + 2 (12 x) + 2 \cdot 4 - 1$

Step 3.5.2

Multiply $12$ by $2$ .

$g (3 x + 2) = 18 x^{2} + 24 x + 2 \cdot 4 - 1$

Step 3.5.3

Multiply $2$ by $4$ .

$g (3 x + 2) = 18 x^{2} + 24 x + 8 - 1$

Step 4

Subtract $1$ from $8$ .

$g (3 x + 2) = 18 x^{2} + 24 x + 7$