Find g(f(x)) f(x)=-x-4 g(x)=-x^2-3x-1

$f (x) = - x - 4$ $g (x) = - x^{2} - 3 x - 1$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (- x - 4)$ by substituting in the value of $f$ into $g$ .

$g (- x - 4) = - {(- x - 4)}^{2} - 3 (- x - 4) - 1$

Step 3

Simplify each term.

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

$g (- x - 4) = - ((- x - 4) (- x - 4)) - 3 (- x - 4) - 1$

Expand $(- x - 4) (- x - 4)$ using the FOIL Method.

Apply the distributive property.

$g (- x - 4) = - (- x (- x - 4) - 4 (- x - 4)) - 3 (- x - 4) - 1$

Apply the distributive property.

$g (- x - 4) = - (- x (- x) - x \cdot - 4 - 4 (- x - 4)) - 3 (- x - 4) - 1$

Apply the distributive property.

$g (- x - 4) = - (- x (- x) - x \cdot - 4 - 4 (- x) - 4 \cdot - 4) - 3 (- x - 4) - 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (- x - 4) = - (- 1 \cdot (- 1 x \cdot x) - x \cdot - 4 - 4 (- x) - 4 \cdot - 4) - 3 (- x - 4) - 1$

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

Move $x$ .

$g (- x - 4) = - (- 1 \cdot (- 1 (x \cdot x)) - x \cdot - 4 - 4 (- x) - 4 \cdot - 4) - 3 (- x - 4) - 1$

Multiply $x$ by $x$ .

$g (- x - 4) = - (- 1 \cdot (- 1 x^{2}) - x \cdot - 4 - 4 (- x) - 4 \cdot - 4) - 3 (- x - 4) - 1$

Multiply $- 1$ by $- 1$ .

$g (- x - 4) = - (1 x^{2} - x \cdot - 4 - 4 (- x) - 4 \cdot - 4) - 3 (- x - 4) - 1$

Multiply $x^{2}$ by $1$ .

$g (- x - 4) = - (x^{2} - x \cdot - 4 - 4 (- x) - 4 \cdot - 4) - 3 (- x - 4) - 1$

Multiply $- 4$ by $- 1$ .

$g (- x - 4) = - (x^{2} + 4 x - 4 (- x) - 4 \cdot - 4) - 3 (- x - 4) - 1$

Multiply $- 1$ by $- 4$ .

$g (- x - 4) = - (x^{2} + 4 x + 4 x - 4 \cdot - 4) - 3 (- x - 4) - 1$

Multiply $- 4$ by $- 4$ .

$g (- x - 4) = - (x^{2} + 4 x + 4 x + 16) - 3 (- x - 4) - 1$

Add $4 x$ and $4 x$ .

$g (- x - 4) = - (x^{2} + 8 x + 16) - 3 (- x - 4) - 1$

Apply the distributive property.

$g (- x - 4) = - x^{2} - (8 x) - 1 \cdot 16 - 3 (- x - 4) - 1$

Simplify.

Multiply $8$ by $- 1$ .

$g (- x - 4) = - x^{2} - 8 x - 1 \cdot 16 - 3 (- x - 4) - 1$

Multiply $- 1$ by $16$ .

$g (- x - 4) = - x^{2} - 8 x - 16 - 3 (- x - 4) - 1$

Apply the distributive property.

$g (- x - 4) = - x^{2} - 8 x - 16 - 3 (- x) - 3 \cdot - 4 - 1$

Multiply $- 1$ by $- 3$ .

$g (- x - 4) = - x^{2} - 8 x - 16 + 3 x - 3 \cdot - 4 - 1$

Multiply $- 3$ by $- 4$ .

$g (- x - 4) = - x^{2} - 8 x - 16 + 3 x + 12 - 1$

Step 4

Simplify by adding terms.

Add $- 8 x$ and $3 x$ .

$g (- x - 4) = - x^{2} - 5 x - 16 + 12 - 1$

Simplify by adding and subtracting.

Add $- 16$ and $12$ .

$g (- x - 4) = - x^{2} - 5 x - 4 - 1$

Subtract $1$ from $- 4$ .

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

Find g(f(x)) f(x)=-x-4 g(x)=-x^2-3x-1

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