Find f(g(x)) f(x)=x^2+9 , g(x)=x^2-1

$f (x) = x^{2} + 9$ , $g (x) = x^{2} - 1$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (x^{2} - 1)$ by substituting in the value of $g$ into $f$ .

$f (x^{2} - 1) = {(x^{2} - 1)}^{2} + 9$

Step 3

Simplify each term.

Rewrite ${(x^{2} - 1)}^{2}$ as $(x^{2} - 1) (x^{2} - 1)$ .

$f (x^{2} - 1) = (x^{2} - 1) (x^{2} - 1) + 9$

Expand $(x^{2} - 1) (x^{2} - 1)$ using the FOIL Method.

Apply the distributive property.

$f (x^{2} - 1) = x^{2} (x^{2} - 1) - 1 (x^{2} - 1) + 9$

Apply the distributive property.

$f (x^{2} - 1) = x^{2} x^{2} + x^{2} \cdot - 1 - 1 (x^{2} - 1) + 9$

Apply the distributive property.

$f (x^{2} - 1) = x^{2} x^{2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1 + 9$

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.

$f (x^{2} - 1) = x^{2 + 2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1 + 9$

Add $2$ and $2$ .

$f (x^{2} - 1) = x^{4} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1 + 9$

Move $- 1$ to the left of $x^{2}$ .

$f (x^{2} - 1) = x^{4} - 1 \cdot x^{2} - 1 x^{2} - 1 \cdot - 1 + 9$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$f (x^{2} - 1) = x^{4} - x^{2} - 1 x^{2} - 1 \cdot - 1 + 9$

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$f (x^{2} - 1) = x^{4} - x^{2} - x^{2} - 1 \cdot - 1 + 9$

Multiply $- 1$ by $- 1$ .

$f (x^{2} - 1) = x^{4} - x^{2} - x^{2} + 1 + 9$

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

$f (x^{2} - 1) = x^{4} - 2 x^{2} + 1 + 9$

Step 4

Add $1$ and $9$ .

$f (x^{2} - 1) = x^{4} - 2 x^{2} + 10$