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

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (4 x - 3)$ by substituting in the value of $g$ into $f$ .

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

Step 3

Simplify each term.

Rewrite ${(4 x - 3)}^{2}$ as $(4 x - 3) (4 x - 3)$ .

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

Expand $(4 x - 3) (4 x - 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $4$ by $4$ .

$f (4 x - 3) = 2 (16 x^{2} + 4 x \cdot - 3 - 3 (4 x) - 3 \cdot - 3) + 2$

Multiply $- 3$ by $4$ .

$f (4 x - 3) = 2 (16 x^{2} - 12 x - 3 (4 x) - 3 \cdot - 3) + 2$

Multiply $4$ by $- 3$ .

$f (4 x - 3) = 2 (16 x^{2} - 12 x - 12 x - 3 \cdot - 3) + 2$

Multiply $- 3$ by $- 3$ .

$f (4 x - 3) = 2 (16 x^{2} - 12 x - 12 x + 9) + 2$

Subtract $12 x$ from $- 12 x$ .

$f (4 x - 3) = 2 (16 x^{2} - 24 x + 9) + 2$

Apply the distributive property.

$f (4 x - 3) = 2 (16 x^{2}) + 2 (- 24 x) + 2 \cdot 9 + 2$

Simplify.

Multiply $16$ by $2$ .

$f (4 x - 3) = 32 x^{2} + 2 (- 24 x) + 2 \cdot 9 + 2$

Multiply $- 24$ by $2$ .

$f (4 x - 3) = 32 x^{2} - 48 x + 2 \cdot 9 + 2$

Multiply $2$ by $9$ .

$f (4 x - 3) = 32 x^{2} - 48 x + 18 + 2$

Step 4

Add $18$ and $2$ .

$f (4 x - 3) = 32 x^{2} - 48 x + 20$

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

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