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

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (3 - 2 x)$ by substituting in the value of $f$ into $g$ .

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

Step 3

Simplify each term.

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

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

Expand $(3 - 2 x) (3 - 2 x)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$g (3 - 2 x) = 3 (9 + 3 (- 2 x) - 2 x \cdot 3 - 2 x (- 2 x)) + 2$

Multiply $- 2$ by $3$ .

$g (3 - 2 x) = 3 (9 - 6 x - 2 x \cdot 3 - 2 x (- 2 x)) + 2$

Multiply $3$ by $- 2$ .

$g (3 - 2 x) = 3 (9 - 6 x - 6 x - 2 x (- 2 x)) + 2$

Rewrite using the commutative property of multiplication.

$g (3 - 2 x) = 3 (9 - 6 x - 6 x - 2 \cdot (- 2 x \cdot x)) + 2$

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

Move $x$ .

$g (3 - 2 x) = 3 (9 - 6 x - 6 x - 2 \cdot (- 2 (x \cdot x))) + 2$

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 2$ .

$g (3 - 2 x) = 3 (9 - 6 x - 6 x + 4 x^{2}) + 2$

Subtract $6 x$ from $- 6 x$ .

$g (3 - 2 x) = 3 (9 - 12 x + 4 x^{2}) + 2$

Apply the distributive property.

$g (3 - 2 x) = 3 \cdot 9 + 3 (- 12 x) + 3 (4 x^{2}) + 2$

Simplify.

Multiply $3$ by $9$ .

$g (3 - 2 x) = 27 + 3 (- 12 x) + 3 (4 x^{2}) + 2$

Multiply $- 12$ by $3$ .

$g (3 - 2 x) = 27 - 36 x + 3 (4 x^{2}) + 2$

Multiply $4$ by $3$ .

$g (3 - 2 x) = 27 - 36 x + 12 x^{2} + 2$

Step 4

Simplify the expression.

Add $27$ and $2$ .

$g (3 - 2 x) = - 36 x + 12 x^{2} + 29$

Reorder $- 36 x$ and $12 x^{2}$ .

$g (3 - 2 x) = 12 x^{2} - 36 x + 29$

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

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