Find f(g(x)) f(x)=x^2-9 , g(x)=4x^2-9

$f (x) = x^{2} - 9$ , $g (x) = 4 x^{2} - 9$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (4 x^{2} - 9)$ by substituting in the value of $g$ into $f$ .

$f (4 x^{2} - 9) = {(4 x^{2} - 9)}^{2} - 9$

Step 3

Simplify each term.

Rewrite ${(4 x^{2} - 9)}^{2}$ as $(4 x^{2} - 9) (4 x^{2} - 9)$ .

$f (4 x^{2} - 9) = (4 x^{2} - 9) (4 x^{2} - 9) - 9$

Expand $(4 x^{2} - 9) (4 x^{2} - 9)$ using the FOIL Method.

Apply the distributive property.

$f (4 x^{2} - 9) = 4 x^{2} (4 x^{2} - 9) - 9 (4 x^{2} - 9) - 9$

Apply the distributive property.

$f (4 x^{2} - 9) = 4 x^{2} (4 x^{2}) + 4 x^{2} \cdot - 9 - 9 (4 x^{2} - 9) - 9$

Apply the distributive property.

$f (4 x^{2} - 9) = 4 x^{2} (4 x^{2}) + 4 x^{2} \cdot - 9 - 9 (4 x^{2}) - 9 \cdot - 9 - 9$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (4 x^{2} - 9) = 4 \cdot (4 x^{2} x^{2}) + 4 x^{2} \cdot - 9 - 9 (4 x^{2}) - 9 \cdot - 9 - 9$

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

Move $x^{2}$ .

$f (4 x^{2} - 9) = 4 \cdot (4 (x^{2} x^{2})) + 4 x^{2} \cdot - 9 - 9 (4 x^{2}) - 9 \cdot - 9 - 9$

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

$f (4 x^{2} - 9) = 4 \cdot (4 x^{2 + 2}) + 4 x^{2} \cdot - 9 - 9 (4 x^{2}) - 9 \cdot - 9 - 9$

Add $2$ and $2$ .

$f (4 x^{2} - 9) = 4 \cdot (4 x^{4}) + 4 x^{2} \cdot - 9 - 9 (4 x^{2}) - 9 \cdot - 9 - 9$

Multiply $4$ by $4$ .

$f (4 x^{2} - 9) = 16 x^{4} + 4 x^{2} \cdot - 9 - 9 (4 x^{2}) - 9 \cdot - 9 - 9$

Multiply $- 9$ by $4$ .

$f (4 x^{2} - 9) = 16 x^{4} - 36 x^{2} - 9 (4 x^{2}) - 9 \cdot - 9 - 9$

Multiply $4$ by $- 9$ .

$f (4 x^{2} - 9) = 16 x^{4} - 36 x^{2} - 36 x^{2} - 9 \cdot - 9 - 9$

Multiply $- 9$ by $- 9$ .

$f (4 x^{2} - 9) = 16 x^{4} - 36 x^{2} - 36 x^{2} + 81 - 9$

Subtract $36 x^{2}$ from $- 36 x^{2}$ .

$f (4 x^{2} - 9) = 16 x^{4} - 72 x^{2} + 81 - 9$

Step 4

Subtract $9$ from $81$ .

$f (4 x^{2} - 9) = 16 x^{4} - 72 x^{2} + 72$