Find g(f(x)) f(x)=x^2+8 , g(x)=x^2-8

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

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

Step 3

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

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.

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

Add $2$ and $2$ .

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

Move $8$ to the left of $x^{2}$ .

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

Multiply $8$ by $8$ .

$g (x^{2} + 8) = x^{4} + 8 x^{2} + 8 x^{2} + 64 - 8$

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

$g (x^{2} + 8) = x^{4} + 16 x^{2} + 64 - 8$

Step 4

Subtract $8$ from $64$ .

$g (x^{2} + 8) = x^{4} + 16 x^{2} + 56$