Find g(f(x)) f(x)=3x-5 , g(x)=4x^2-2

$f (x) = 3 x - 5$ , $g (x) = 4 x^{2} - 2$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (3 x - 5)$ by substituting in the value of $f$ into $g$ .

$g (3 x - 5) = 4 {(3 x - 5)}^{2} - 2$

Step 3

Simplify each term.

Rewrite ${(3 x - 5)}^{2}$ as $(3 x - 5) (3 x - 5)$ .

$g (3 x - 5) = 4 ((3 x - 5) (3 x - 5)) - 2$

Expand $(3 x - 5) (3 x - 5)$ using the FOIL Method.

Apply the distributive property.

$g (3 x - 5) = 4 (3 x (3 x - 5) - 5 (3 x - 5)) - 2$

Apply the distributive property.

$g (3 x - 5) = 4 (3 x (3 x) + 3 x \cdot - 5 - 5 (3 x - 5)) - 2$

Apply the distributive property.

$g (3 x - 5) = 4 (3 x (3 x) + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5) - 2$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (3 x - 5) = 4 (3 \cdot (3 x \cdot x) + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5) - 2$

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

Move $x$ .

$g (3 x - 5) = 4 (3 \cdot (3 (x \cdot x)) + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5) - 2$

Multiply $x$ by $x$ .

$g (3 x - 5) = 4 (3 \cdot (3 x^{2}) + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5) - 2$

Multiply $3$ by $3$ .

$g (3 x - 5) = 4 (9 x^{2} + 3 x \cdot - 5 - 5 (3 x) - 5 \cdot - 5) - 2$

Multiply $- 5$ by $3$ .

$g (3 x - 5) = 4 (9 x^{2} - 15 x - 5 (3 x) - 5 \cdot - 5) - 2$

Multiply $3$ by $- 5$ .

$g (3 x - 5) = 4 (9 x^{2} - 15 x - 15 x - 5 \cdot - 5) - 2$

Multiply $- 5$ by $- 5$ .

$g (3 x - 5) = 4 (9 x^{2} - 15 x - 15 x + 25) - 2$

Subtract $15 x$ from $- 15 x$ .

$g (3 x - 5) = 4 (9 x^{2} - 30 x + 25) - 2$

Apply the distributive property.

$g (3 x - 5) = 4 (9 x^{2}) + 4 (- 30 x) + 4 \cdot 25 - 2$

Simplify.

Multiply $9$ by $4$ .

$g (3 x - 5) = 36 x^{2} + 4 (- 30 x) + 4 \cdot 25 - 2$

Multiply $- 30$ by $4$ .

$g (3 x - 5) = 36 x^{2} - 120 x + 4 \cdot 25 - 2$

Multiply $4$ by $25$ .

$g (3 x - 5) = 36 x^{2} - 120 x + 100 - 2$

Step 4

Subtract $2$ from $100$ .

$g (3 x - 5) = 36 x^{2} - 120 x + 98$

Find g(f(x)) f(x)=3x-5 , g(x)=4x^2-2

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