Find g(f(x)) f(x)=x^2+2 , g(x)=x^2-2

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

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

Step 3

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

Add $2$ and $2$ .

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

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

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

Multiply $2$ by $2$ .

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

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

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

Step 4

Subtract $2$ from $4$ .

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