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

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (1 - x)$ by substituting in the value of $f$ into $g$ .

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

Step 3

Remove parentheses.

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

Step 4

Simplify each term.

Rewrite ${(1 - x)}^{2}$ as $(1 - x) (1 - x)$ .

$g (1 - x) = 3 ((1 - x) (1 - x)) + 1 - x + 2$

Expand $(1 - x) (1 - x)$ using the FOIL Method.

Apply the distributive property.

$g (1 - x) = 3 (1 (1 - x) - x (1 - x)) + 1 - x + 2$

Apply the distributive property.

$g (1 - x) = 3 (1 \cdot 1 + 1 (- x) - x (1 - x)) + 1 - x + 2$

Apply the distributive property.

$g (1 - x) = 3 (1 \cdot 1 + 1 (- x) - x \cdot 1 - x (- x)) + 1 - x + 2$

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

$g (1 - x) = 3 (1 + 1 (- x) - x \cdot 1 - x (- x)) + 1 - x + 2$

Multiply $- x$ by $1$ .

$g (1 - x) = 3 (1 - x - x \cdot 1 - x (- x)) + 1 - x + 2$

Multiply $- 1$ by $1$ .

$g (1 - x) = 3 (1 - x - x - x (- x)) + 1 - x + 2$

Rewrite using the commutative property of multiplication.

$g (1 - x) = 3 (1 - x - x - 1 \cdot (- 1 x \cdot x)) + 1 - x + 2$

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

Move $x$ .

$g (1 - x) = 3 (1 - x - x - 1 \cdot (- 1 (x \cdot x))) + 1 - x + 2$

Multiply $x$ by $x$ .

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

Multiply $- 1$ by $- 1$ .

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

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

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

Subtract $x$ from $- x$ .

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

Apply the distributive property.

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

Simplify.

Multiply $3$ by $1$ .

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

Multiply $- 2$ by $3$ .

$g (1 - x) = 3 - 6 x + 3 x^{2} + 1 - x + 2$

Step 5

Simplify by adding terms.

Add $3$ and $1$ .

$g (1 - x) = - 6 x + 3 x^{2} + 4 - x + 2$

Subtract $x$ from $- 6 x$ .

$g (1 - x) = - 7 x + 3 x^{2} + 4 + 2$

Simplify the expression.

Add $4$ and $2$ .

$g (1 - x) = - 7 x + 3 x^{2} + 6$

Reorder $- 7 x$ and $3 x^{2}$ .

$g (1 - x) = 3 x^{2} - 7 x + 6$

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

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