Find f(g(x)) f(x)=x^2-4x-9 g(x)=-3x-5

$f (x) = x^{2} - 4 x - 9$ $g (x) = - 3 x - 5$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (- 3 x - 5)$ by substituting in the value of $g$ into $f$ .

$f (- 3 x - 5) = {(- 3 x - 5)}^{2} - 4 (- 3 x - 5) - 9$

Step 3

Simplify each term.

Rewrite ${(- 3 x - 5)}^{2}$ as $(- 3 x - 5) (- 3 x - 5)$ .

$f (- 3 x - 5) = (- 3 x - 5) (- 3 x - 5) - 4 (- 3 x - 5) - 9$

Expand $(- 3 x - 5) (- 3 x - 5)$ using the FOIL Method.

Apply the distributive property.

$f (- 3 x - 5) = - 3 x (- 3 x - 5) - 5 (- 3 x - 5) - 4 (- 3 x - 5) - 9$

Apply the distributive property.

$f (- 3 x - 5) = - 3 x (- 3 x) - 3 x \cdot - 5 - 5 (- 3 x - 5) - 4 (- 3 x - 5) - 9$

Apply the distributive property.

$f (- 3 x - 5) = - 3 x (- 3 x) - 3 x \cdot - 5 - 5 (- 3 x) - 5 \cdot - 5 - 4 (- 3 x - 5) - 9$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (- 3 x - 5) = - 3 \cdot (- 3 x \cdot x) - 3 x \cdot - 5 - 5 (- 3 x) - 5 \cdot - 5 - 4 (- 3 x - 5) - 9$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$f (- 3 x - 5) = - 3 \cdot (- 3 (x \cdot x)) - 3 x \cdot - 5 - 5 (- 3 x) - 5 \cdot - 5 - 4 (- 3 x - 5) - 9$

Multiply $x$ by $x$ .

$f (- 3 x - 5) = - 3 \cdot (- 3 x^{2}) - 3 x \cdot - 5 - 5 (- 3 x) - 5 \cdot - 5 - 4 (- 3 x - 5) - 9$

Multiply $- 3$ by $- 3$ .

$f (- 3 x - 5) = 9 x^{2} - 3 x \cdot - 5 - 5 (- 3 x) - 5 \cdot - 5 - 4 (- 3 x - 5) - 9$

Multiply $- 5$ by $- 3$ .

$f (- 3 x - 5) = 9 x^{2} + 15 x - 5 (- 3 x) - 5 \cdot - 5 - 4 (- 3 x - 5) - 9$

Multiply $- 3$ by $- 5$ .

$f (- 3 x - 5) = 9 x^{2} + 15 x + 15 x - 5 \cdot - 5 - 4 (- 3 x - 5) - 9$

Multiply $- 5$ by $- 5$ .

$f (- 3 x - 5) = 9 x^{2} + 15 x + 15 x + 25 - 4 (- 3 x - 5) - 9$

Add $15 x$ and $15 x$ .

$f (- 3 x - 5) = 9 x^{2} + 30 x + 25 - 4 (- 3 x - 5) - 9$

Apply the distributive property.

$f (- 3 x - 5) = 9 x^{2} + 30 x + 25 - 4 (- 3 x) - 4 \cdot - 5 - 9$

Multiply $- 3$ by $- 4$ .

$f (- 3 x - 5) = 9 x^{2} + 30 x + 25 + 12 x - 4 \cdot - 5 - 9$

Multiply $- 4$ by $- 5$ .

$f (- 3 x - 5) = 9 x^{2} + 30 x + 25 + 12 x + 20 - 9$

Step 4

Simplify by adding terms.

Add $30 x$ and $12 x$ .

$f (- 3 x - 5) = 9 x^{2} + 42 x + 25 + 20 - 9$

Simplify by adding and subtracting.

Add $25$ and $20$ .

$f (- 3 x - 5) = 9 x^{2} + 42 x + 45 - 9$

Subtract $9$ from $45$ .

$f (- 3 x - 5) = 9 x^{2} + 42 x + 36$

Find f(g(x)) f(x)=x^2-4x-9 g(x)=-3x-5

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