Find g(f(x)) f(x)=x+3 , g(x)=2x^2-5x-3

$f (x) = x + 3$ , $g (x) = 2 x^{2} - 5 x - 3$

Set up the composite result function.

$g (f (x))$

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

$g (x + 3) = 2 {(x + 3)}^{2} - 5 (x + 3) - 3$

Simplify each term.

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

$g (x + 3) = 2 ((x + 3) (x + 3)) - 5 (x + 3) - 3$

Expand $(x + 3) (x + 3)$ using the FOIL Method.

Apply the distributive property.

$g (x + 3) = 2 (x (x + 3) + 3 (x + 3)) - 5 (x + 3) - 3$

Apply the distributive property.

$g (x + 3) = 2 (x \cdot x + x \cdot 3 + 3 (x + 3)) - 5 (x + 3) - 3$

Apply the distributive property.

$g (x + 3) = 2 (x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3) - 5 (x + 3) - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$g (x + 3) = 2 (x^{2} + x \cdot 3 + 3 x + 3 \cdot 3) - 5 (x + 3) - 3$

Move $3$ to the left of $x$ .

$g (x + 3) = 2 (x^{2} + 3 \cdot x + 3 x + 3 \cdot 3) - 5 (x + 3) - 3$

Multiply $3$ by $3$ .

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

Add $3 x$ and $3 x$ .

$g (x + 3) = 2 (x^{2} + 6 x + 9) - 5 (x + 3) - 3$

Apply the distributive property.

$g (x + 3) = 2 x^{2} + 2 (6 x) + 2 \cdot 9 - 5 (x + 3) - 3$

Simplify.

Multiply $6$ by $2$ .

$g (x + 3) = 2 x^{2} + 12 x + 2 \cdot 9 - 5 (x + 3) - 3$

Multiply $2$ by $9$ .

$g (x + 3) = 2 x^{2} + 12 x + 18 - 5 (x + 3) - 3$

Apply the distributive property.

$g (x + 3) = 2 x^{2} + 12 x + 18 - 5 x - 5 \cdot 3 - 3$

Multiply $- 5$ by $3$ .

$g (x + 3) = 2 x^{2} + 12 x + 18 - 5 x - 15 - 3$

Simplify by adding terms.

Subtract $5 x$ from $12 x$ .

$g (x + 3) = 2 x^{2} + 7 x + 18 - 15 - 3$

Subtract $15$ from $18$ .

$g (x + 3) = 2 x^{2} + 7 x + 3 - 3$

Combine the opposite terms in $2 x^{2} + 7 x + 3 - 3$ .

Subtract $3$ from $3$ .

$g (x + 3) = 2 x^{2} + 7 x + 0$

Add $2 x^{2} + 7 x$ and $0$ .

$g (x + 3) = 2 x^{2} + 7 x$

Find g(f(x)) f(x)=x+3 , g(x)=2x^2-5x-3

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