Find f(g(x)) f(x)=2x^2-8x+4 and g(x)=3x+7

$f (x) = 2 x^{2} - 8 x + 4$ and $g (x) = 3 x + 7$

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (3 x + 7)$ by substituting in the value of $g$ into $f$ .

$f (3 x + 7) = 2 {(3 x + 7)}^{2} - 8 (3 x + 7) + 4$

Step 3

Simplify each term.

Rewrite ${(3 x + 7)}^{2}$ as $(3 x + 7) (3 x + 7)$ .

$f (3 x + 7) = 2 ((3 x + 7) (3 x + 7)) - 8 (3 x + 7) + 4$

Expand $(3 x + 7) (3 x + 7)$ using the FOIL Method.

Apply the distributive property.

$f (3 x + 7) = 2 (3 x (3 x + 7) + 7 (3 x + 7)) - 8 (3 x + 7) + 4$

Apply the distributive property.

$f (3 x + 7) = 2 (3 x (3 x) + 3 x \cdot 7 + 7 (3 x + 7)) - 8 (3 x + 7) + 4$

Apply the distributive property.

$f (3 x + 7) = 2 (3 x (3 x) + 3 x \cdot 7 + 7 (3 x) + 7 \cdot 7) - 8 (3 x + 7) + 4$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$f (3 x + 7) = 2 (3 \cdot (3 x \cdot x) + 3 x \cdot 7 + 7 (3 x) + 7 \cdot 7) - 8 (3 x + 7) + 4$

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

Move $x$ .

$f (3 x + 7) = 2 (3 \cdot (3 (x \cdot x)) + 3 x \cdot 7 + 7 (3 x) + 7 \cdot 7) - 8 (3 x + 7) + 4$

Multiply $x$ by $x$ .

$f (3 x + 7) = 2 (3 \cdot (3 x^{2}) + 3 x \cdot 7 + 7 (3 x) + 7 \cdot 7) - 8 (3 x + 7) + 4$

Multiply $3$ by $3$ .

$f (3 x + 7) = 2 (9 x^{2} + 3 x \cdot 7 + 7 (3 x) + 7 \cdot 7) - 8 (3 x + 7) + 4$

Multiply $7$ by $3$ .

$f (3 x + 7) = 2 (9 x^{2} + 21 x + 7 (3 x) + 7 \cdot 7) - 8 (3 x + 7) + 4$

Multiply $3$ by $7$ .

$f (3 x + 7) = 2 (9 x^{2} + 21 x + 21 x + 7 \cdot 7) - 8 (3 x + 7) + 4$

Multiply $7$ by $7$ .

$f (3 x + 7) = 2 (9 x^{2} + 21 x + 21 x + 49) - 8 (3 x + 7) + 4$

Add $21 x$ and $21 x$ .

$f (3 x + 7) = 2 (9 x^{2} + 42 x + 49) - 8 (3 x + 7) + 4$

Apply the distributive property.

$f (3 x + 7) = 2 (9 x^{2}) + 2 (42 x) + 2 \cdot 49 - 8 (3 x + 7) + 4$

Simplify.

Multiply $9$ by $2$ .

$f (3 x + 7) = 18 x^{2} + 2 (42 x) + 2 \cdot 49 - 8 (3 x + 7) + 4$

Multiply $42$ by $2$ .

$f (3 x + 7) = 18 x^{2} + 84 x + 2 \cdot 49 - 8 (3 x + 7) + 4$

Multiply $2$ by $49$ .

$f (3 x + 7) = 18 x^{2} + 84 x + 98 - 8 (3 x + 7) + 4$

Apply the distributive property.

$f (3 x + 7) = 18 x^{2} + 84 x + 98 - 8 (3 x) - 8 \cdot 7 + 4$

Multiply $3$ by $- 8$ .

$f (3 x + 7) = 18 x^{2} + 84 x + 98 - 24 x - 8 \cdot 7 + 4$

Multiply $- 8$ by $7$ .

$f (3 x + 7) = 18 x^{2} + 84 x + 98 - 24 x - 56 + 4$

Step 4

Simplify by adding terms.

Subtract $24 x$ from $84 x$ .

$f (3 x + 7) = 18 x^{2} + 60 x + 98 - 56 + 4$

Simplify by adding and subtracting.

Subtract $56$ from $98$ .

$f (3 x + 7) = 18 x^{2} + 60 x + 42 + 4$

Add $42$ and $4$ .

$f (3 x + 7) = 18 x^{2} + 60 x + 46$

Find f(g(x)) f(x)=2x^2-8x+4 and g(x)=3x+7

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