Find g(f(x)) f(x)=x+2 , g(x)=2x^2-3x-2

$f (x) = x + 2$ , $g (x) = 2 x^{2} - 3 x - 2$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (x + 2)$ by substituting in the value of $f$ into $g$ .

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

Step 3

Simplify each term.

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

$g (x + 2) = 2 ((x + 2) (x + 2)) - 3 (x + 2) - 2$

Expand $(x + 2) (x + 2)$ using the FOIL Method.

Apply the distributive property.

$g (x + 2) = 2 (x (x + 2) + 2 (x + 2)) - 3 (x + 2) - 2$

Apply the distributive property.

$g (x + 2) = 2 (x \cdot x + x \cdot 2 + 2 (x + 2)) - 3 (x + 2) - 2$

Apply the distributive property.

$g (x + 2) = 2 (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) - 3 (x + 2) - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $2$ to the left of $x$ .

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

Multiply $2$ by $2$ .

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

Add $2 x$ and $2 x$ .

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

Apply the distributive property.

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

Simplify.

Multiply $4$ by $2$ .

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

Multiply $2$ by $4$ .

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

Apply the distributive property.

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

Multiply $- 3$ by $2$ .

$g (x + 2) = 2 x^{2} + 8 x + 8 - 3 x - 6 - 2$

Step 4

Simplify by adding terms.

Subtract $3 x$ from $8 x$ .

$g (x + 2) = 2 x^{2} + 5 x + 8 - 6 - 2$

Subtract $6$ from $8$ .

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

Combine the opposite terms in $2 x^{2} + 5 x + 2 - 2$ .

Subtract $2$ from $2$ .

$g (x + 2) = 2 x^{2} + 5 x + 0$

Add $2 x^{2} + 5 x$ and $0$ .

$g (x + 2) = 2 x^{2} + 5 x$

Find g(f(x)) f(x)=x+2 , g(x)=2x^2-3x-2

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