Find f(g(x)) f(x)=x^2+2x-8 g(x)=-x^2+4

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

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 $f$ .

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

Step 3

Simplify each term.

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (- x^{2} + 4) = - 1 \cdot (- 1 x^{2} x^{2}) - x^{2} \cdot 4 + 4 (- x^{2}) + 4 \cdot 4 + 2 (- x^{2} + 4) - 8$

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

Move $x^{2}$ .

$f (- x^{2} + 4) = - 1 \cdot (- 1 (x^{2} x^{2})) - x^{2} \cdot 4 + 4 (- x^{2}) + 4 \cdot 4 + 2 (- x^{2} + 4) - 8$

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

$f (- x^{2} + 4) = - 1 \cdot (- 1 x^{2 + 2}) - x^{2} \cdot 4 + 4 (- x^{2}) + 4 \cdot 4 + 2 (- x^{2} + 4) - 8$

Add $2$ and $2$ .

$f (- x^{2} + 4) = - 1 \cdot (- 1 x^{4}) - x^{2} \cdot 4 + 4 (- x^{2}) + 4 \cdot 4 + 2 (- x^{2} + 4) - 8$

Multiply $- 1$ by $- 1$ .

$f (- x^{2} + 4) = 1 x^{4} - x^{2} \cdot 4 + 4 (- x^{2}) + 4 \cdot 4 + 2 (- x^{2} + 4) - 8$

Multiply $x^{4}$ by $1$ .

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

Multiply $4$ by $- 1$ .

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

Multiply $- 1$ by $4$ .

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

Multiply $4$ by $4$ .

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $2$ .

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

Multiply $2$ by $4$ .

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

Step 4

Simplify by adding terms.

Combine the opposite terms in $x^{4} - 8 x^{2} + 16 - 2 x^{2} + 8 - 8$ .

Subtract $8$ from $8$ .

$f (- x^{2} + 4) = x^{4} - 8 x^{2} + 16 - 2 x^{2} + 0$

Add $x^{4} - 8 x^{2} + 16 - 2 x^{2}$ and $0$ .

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

Subtract $2 x^{2}$ from $- 8 x^{2}$ .

$f (- x^{2} + 4) = x^{4} - 10 x^{2} + 16$

Find f(g(x)) f(x)=x^2+2x-8 g(x)=-x^2+4

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