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

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

Step 1

Set up the composite result function.

$f (g (x))$

Step 2

Evaluate $f (3 x - 2)$ by substituting in the value of $g$ into $f$ .

$f (3 x - 2) = 3 {(3 x - 2)}^{2} + 5$

Step 3

Simplify each term.

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

$f (3 x - 2) = 3 ((3 x - 2) (3 x - 2)) + 5$

Expand $(3 x - 2) (3 x - 2)$ using the FOIL Method.

Apply the distributive property.

$f (3 x - 2) = 3 (3 x (3 x - 2) - 2 (3 x - 2)) + 5$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $3$ by $3$ .

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

Multiply $- 2$ by $3$ .

$f (3 x - 2) = 3 (9 x^{2} - 6 x - 2 (3 x) - 2 \cdot - 2) + 5$

Multiply $3$ by $- 2$ .

$f (3 x - 2) = 3 (9 x^{2} - 6 x - 6 x - 2 \cdot - 2) + 5$

Multiply $- 2$ by $- 2$ .

$f (3 x - 2) = 3 (9 x^{2} - 6 x - 6 x + 4) + 5$

Subtract $6 x$ from $- 6 x$ .

$f (3 x - 2) = 3 (9 x^{2} - 12 x + 4) + 5$

Apply the distributive property.

$f (3 x - 2) = 3 (9 x^{2}) + 3 (- 12 x) + 3 \cdot 4 + 5$

Simplify.

Multiply $9$ by $3$ .

$f (3 x - 2) = 27 x^{2} + 3 (- 12 x) + 3 \cdot 4 + 5$

Multiply $- 12$ by $3$ .

$f (3 x - 2) = 27 x^{2} - 36 x + 3 \cdot 4 + 5$

Multiply $3$ by $4$ .

$f (3 x - 2) = 27 x^{2} - 36 x + 12 + 5$

Step 4

Add $12$ and $5$ .

$f (3 x - 2) = 27 x^{2} - 36 x + 17$

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

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