Find g(f(x)) f(x)=3x-4 , g(x)=2x^2-1

$f (x) = 3 x - 4$ , $g (x) = 2 x^{2} - 1$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

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

$g (3 x - 4) = 2 {(3 x - 4)}^{2} - 1$

Step 3

Simplify each term.

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

$g (3 x - 4) = 2 ((3 x - 4) (3 x - 4)) - 1$

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

Apply the distributive property.

$g (3 x - 4) = 2 (3 x (3 x - 4) - 4 (3 x - 4)) - 1$

Apply the distributive property.

$g (3 x - 4) = 2 (3 x (3 x) + 3 x \cdot - 4 - 4 (3 x - 4)) - 1$

Apply the distributive property.

$g (3 x - 4) = 2 (3 x (3 x) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4) - 1$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (3 x - 4) = 2 (3 \cdot (3 x \cdot x) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4) - 1$

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

Move $x$ .

$g (3 x - 4) = 2 (3 \cdot (3 (x \cdot x)) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4) - 1$

Multiply $x$ by $x$ .

$g (3 x - 4) = 2 (3 \cdot (3 x^{2}) + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4) - 1$

Multiply $3$ by $3$ .

$g (3 x - 4) = 2 (9 x^{2} + 3 x \cdot - 4 - 4 (3 x) - 4 \cdot - 4) - 1$

Multiply $- 4$ by $3$ .

$g (3 x - 4) = 2 (9 x^{2} - 12 x - 4 (3 x) - 4 \cdot - 4) - 1$

Multiply $3$ by $- 4$ .

$g (3 x - 4) = 2 (9 x^{2} - 12 x - 12 x - 4 \cdot - 4) - 1$

Multiply $- 4$ by $- 4$ .

$g (3 x - 4) = 2 (9 x^{2} - 12 x - 12 x + 16) - 1$

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

$g (3 x - 4) = 2 (9 x^{2} - 24 x + 16) - 1$

Apply the distributive property.

$g (3 x - 4) = 2 (9 x^{2}) + 2 (- 24 x) + 2 \cdot 16 - 1$

Simplify.

Multiply $9$ by $2$ .

$g (3 x - 4) = 18 x^{2} + 2 (- 24 x) + 2 \cdot 16 - 1$

Multiply $- 24$ by $2$ .

$g (3 x - 4) = 18 x^{2} - 48 x + 2 \cdot 16 - 1$

Multiply $2$ by $16$ .

$g (3 x - 4) = 18 x^{2} - 48 x + 32 - 1$

Step 4

Subtract $1$ from $32$ .

$g (3 x - 4) = 18 x^{2} - 48 x + 31$

Find g(f(x)) f(x)=3x-4 , g(x)=2x^2-1

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