Finite Math Examples

$f (x) = x^{2} - 4$ , $g (x) = x^{2} + 2 x$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (x^{2} - 4)$ by substituting in the value of $f$ into $g$ .

$g (x^{2} - 4) = {(x^{2} - 4)}^{2} + 2 (x^{2} - 4)$

Step 3

Simplify each term.

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Step 3.1

Rewrite ${(x^{2} - 4)}^{2}$ as $(x^{2} - 4) (x^{2} - 4)$ .

$g (x^{2} - 4) = (x^{2} - 4) (x^{2} - 4) + 2 (x^{2} - 4)$

Step 3.2

Expand $(x^{2} - 4) (x^{2} - 4)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$g (x^{2} - 4) = x^{2} (x^{2} - 4) - 4 (x^{2} - 4) + 2 (x^{2} - 4)$

Step 3.2.2

Apply the distributive property.

$g (x^{2} - 4) = x^{2} x^{2} + x^{2} \cdot - 4 - 4 (x^{2} - 4) + 2 (x^{2} - 4)$

Step 3.2.3

Apply the distributive property.

$g (x^{2} - 4) = x^{2} x^{2} + x^{2} \cdot - 4 - 4 x^{2} - 4 \cdot - 4 + 2 (x^{2} - 4)$

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

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

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Step 3.3.1.1.1

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

$g (x^{2} - 4) = x^{2 + 2} + x^{2} \cdot - 4 - 4 x^{2} - 4 \cdot - 4 + 2 (x^{2} - 4)$

Step 3.3.1.1.2

Add $2$ and $2$ .

$g (x^{2} - 4) = x^{4} + x^{2} \cdot - 4 - 4 x^{2} - 4 \cdot - 4 + 2 (x^{2} - 4)$

Step 3.3.1.2

Move $- 4$ to the left of $x^{2}$ .

$g (x^{2} - 4) = x^{4} - 4 \cdot x^{2} - 4 x^{2} - 4 \cdot - 4 + 2 (x^{2} - 4)$

Step 3.3.1.3

Multiply $- 4$ by $- 4$ .

$g (x^{2} - 4) = x^{4} - 4 x^{2} - 4 x^{2} + 16 + 2 (x^{2} - 4)$

Step 3.3.2

Subtract $4 x^{2}$ from $- 4 x^{2}$ .

$g (x^{2} - 4) = x^{4} - 8 x^{2} + 16 + 2 (x^{2} - 4)$

Step 3.4

Apply the distributive property.

$g (x^{2} - 4) = x^{4} - 8 x^{2} + 16 + 2 x^{2} + 2 \cdot - 4$

Step 3.5

Multiply $2$ by $- 4$ .

$g (x^{2} - 4) = x^{4} - 8 x^{2} + 16 + 2 x^{2} - 8$

Step 4

Simplify by adding terms.

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Step 4.1

Add $- 8 x^{2}$ and $2 x^{2}$ .

$g (x^{2} - 4) = x^{4} - 6 x^{2} + 16 - 8$

Step 4.2

Subtract $8$ from $16$ .

$g (x^{2} - 4) = x^{4} - 6 x^{2} + 8$