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

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

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (- 2 x + 4)$ by substituting in the value of $f$ into $g$ .

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

Step 3

Remove parentheses.

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

Step 4

Simplify each term.

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

$g (- 2 x + 4) = (- 2 x + 4) (- 2 x + 4) - 2 x + 4 - 1$

Expand $(- 2 x + 4) (- 2 x + 4)$ using the FOIL Method.

Apply the distributive property.

$g (- 2 x + 4) = - 2 x (- 2 x + 4) + 4 (- 2 x + 4) - 2 x + 4 - 1$

Apply the distributive property.

$g (- 2 x + 4) = - 2 x (- 2 x) - 2 x \cdot 4 + 4 (- 2 x + 4) - 2 x + 4 - 1$

Apply the distributive property.

$g (- 2 x + 4) = - 2 x (- 2 x) - 2 x \cdot 4 + 4 (- 2 x) + 4 \cdot 4 - 2 x + 4 - 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (- 2 x + 4) = - 2 \cdot (- 2 x \cdot x) - 2 x \cdot 4 + 4 (- 2 x) + 4 \cdot 4 - 2 x + 4 - 1$

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

Move $x$ .

$g (- 2 x + 4) = - 2 \cdot (- 2 (x \cdot x)) - 2 x \cdot 4 + 4 (- 2 x) + 4 \cdot 4 - 2 x + 4 - 1$

Multiply $x$ by $x$ .

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

Multiply $- 2$ by $- 2$ .

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

Multiply $4$ by $- 2$ .

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

Multiply $- 2$ by $4$ .

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

Multiply $4$ by $4$ .

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

Subtract $8 x$ from $- 8 x$ .

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

Step 5

Simplify by adding terms.

Subtract $2 x$ from $- 16 x$ .

$g (- 2 x + 4) = 4 x^{2} - 18 x + 16 + 4 - 1$

Simplify by adding and subtracting.

Add $16$ and $4$ .

$g (- 2 x + 4) = 4 x^{2} - 18 x + 20 - 1$

Subtract $1$ from $20$ .

$g (- 2 x + 4) = 4 x^{2} - 18 x + 19$

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

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