Find f(g(x)) f(x)=4x^2+3x+4 , g(x)=3x-6

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (3 x - 6)$ by substituting in the value of $g$ into $f$ .

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

Step 3

Simplify each term.

Rewrite ${(3 x - 6)}^{2}$ as $(3 x - 6) (3 x - 6)$ .

$f (3 x - 6) = 4 ((3 x - 6) (3 x - 6)) + 3 (3 x - 6) + 4$

Expand $(3 x - 6) (3 x - 6)$ using the FOIL Method.

Apply the distributive property.

$f (3 x - 6) = 4 (3 x (3 x - 6) - 6 (3 x - 6)) + 3 (3 x - 6) + 4$

Apply the distributive property.

$f (3 x - 6) = 4 (3 x (3 x) + 3 x \cdot - 6 - 6 (3 x - 6)) + 3 (3 x - 6) + 4$

Apply the distributive property.

$f (3 x - 6) = 4 (3 x (3 x) + 3 x \cdot - 6 - 6 (3 x) - 6 \cdot - 6) + 3 (3 x - 6) + 4$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (3 x - 6) = 4 (3 \cdot (3 x \cdot x) + 3 x \cdot - 6 - 6 (3 x) - 6 \cdot - 6) + 3 (3 x - 6) + 4$

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

Move $x$ .

$f (3 x - 6) = 4 (3 \cdot (3 (x \cdot x)) + 3 x \cdot - 6 - 6 (3 x) - 6 \cdot - 6) + 3 (3 x - 6) + 4$

Multiply $x$ by $x$ .

$f (3 x - 6) = 4 (3 \cdot (3 x^{2}) + 3 x \cdot - 6 - 6 (3 x) - 6 \cdot - 6) + 3 (3 x - 6) + 4$

Multiply $3$ by $3$ .

$f (3 x - 6) = 4 (9 x^{2} + 3 x \cdot - 6 - 6 (3 x) - 6 \cdot - 6) + 3 (3 x - 6) + 4$

Multiply $- 6$ by $3$ .

$f (3 x - 6) = 4 (9 x^{2} - 18 x - 6 (3 x) - 6 \cdot - 6) + 3 (3 x - 6) + 4$

Multiply $3$ by $- 6$ .

$f (3 x - 6) = 4 (9 x^{2} - 18 x - 18 x - 6 \cdot - 6) + 3 (3 x - 6) + 4$

Multiply $- 6$ by $- 6$ .

$f (3 x - 6) = 4 (9 x^{2} - 18 x - 18 x + 36) + 3 (3 x - 6) + 4$

Subtract $18 x$ from $- 18 x$ .

$f (3 x - 6) = 4 (9 x^{2} - 36 x + 36) + 3 (3 x - 6) + 4$

Apply the distributive property.

$f (3 x - 6) = 4 (9 x^{2}) + 4 (- 36 x) + 4 \cdot 36 + 3 (3 x - 6) + 4$

Simplify.

Multiply $9$ by $4$ .

$f (3 x - 6) = 36 x^{2} + 4 (- 36 x) + 4 \cdot 36 + 3 (3 x - 6) + 4$

Multiply $- 36$ by $4$ .

$f (3 x - 6) = 36 x^{2} - 144 x + 4 \cdot 36 + 3 (3 x - 6) + 4$

Multiply $4$ by $36$ .

$f (3 x - 6) = 36 x^{2} - 144 x + 144 + 3 (3 x - 6) + 4$

Apply the distributive property.

$f (3 x - 6) = 36 x^{2} - 144 x + 144 + 3 (3 x) + 3 \cdot - 6 + 4$

Multiply $3$ by $3$ .

$f (3 x - 6) = 36 x^{2} - 144 x + 144 + 9 x + 3 \cdot - 6 + 4$

Multiply $3$ by $- 6$ .

$f (3 x - 6) = 36 x^{2} - 144 x + 144 + 9 x - 18 + 4$

Step 4

Simplify by adding terms.

Add $- 144 x$ and $9 x$ .

$f (3 x - 6) = 36 x^{2} - 135 x + 144 - 18 + 4$

Simplify by adding and subtracting.

Subtract $18$ from $144$ .

$f (3 x - 6) = 36 x^{2} - 135 x + 126 + 4$

Add $126$ and $4$ .

$f (3 x - 6) = 36 x^{2} - 135 x + 130$

Find f(g(x)) f(x)=4x^2+3x+4 , g(x)=3x-6

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