Find f(g(x)) f(x)=x^2+4 , g(x)=x^2-4

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

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

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

Step 3

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

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.

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

Add $2$ and $2$ .

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

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

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

Multiply $- 4$ by $- 4$ .

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

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

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

Step 4

Add $16$ and $4$ .

$f (x^{2} - 4) = x^{4} - 8 x^{2} + 20$