Find g(f(x)) f(x)=14x+19 g(x)=-13x^2+7x+30

$f (x) = 14 x + 19$ $g (x) = - 13 x^{2} + 7 x + 30$

Step 1

Set up the composite result function.

$g (f (x))$

Step 2

Evaluate $g (14 x + 19)$ by substituting in the value of $f$ into $g$ .

$g (14 x + 19) = - 13 {(14 x + 19)}^{2} + 7 (14 x + 19) + 30$

Step 3

Simplify each term.

Rewrite ${(14 x + 19)}^{2}$ as $(14 x + 19) (14 x + 19)$ .

$g (14 x + 19) = - 13 ((14 x + 19) (14 x + 19)) + 7 (14 x + 19) + 30$

Expand $(14 x + 19) (14 x + 19)$ using the FOIL Method.

Apply the distributive property.

$g (14 x + 19) = - 13 (14 x (14 x + 19) + 19 (14 x + 19)) + 7 (14 x + 19) + 30$

Apply the distributive property.

$g (14 x + 19) = - 13 (14 x (14 x) + 14 x \cdot 19 + 19 (14 x + 19)) + 7 (14 x + 19) + 30$

Apply the distributive property.

$g (14 x + 19) = - 13 (14 x (14 x) + 14 x \cdot 19 + 19 (14 x) + 19 \cdot 19) + 7 (14 x + 19) + 30$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$g (14 x + 19) = - 13 (14 \cdot (14 x \cdot x) + 14 x \cdot 19 + 19 (14 x) + 19 \cdot 19) + 7 (14 x + 19) + 30$

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

Move $x$ .

$g (14 x + 19) = - 13 (14 \cdot (14 (x \cdot x)) + 14 x \cdot 19 + 19 (14 x) + 19 \cdot 19) + 7 (14 x + 19) + 30$

Multiply $x$ by $x$ .

$g (14 x + 19) = - 13 (14 \cdot (14 x^{2}) + 14 x \cdot 19 + 19 (14 x) + 19 \cdot 19) + 7 (14 x + 19) + 30$

Multiply $14$ by $14$ .

$g (14 x + 19) = - 13 (196 x^{2} + 14 x \cdot 19 + 19 (14 x) + 19 \cdot 19) + 7 (14 x + 19) + 30$

Multiply $19$ by $14$ .

$g (14 x + 19) = - 13 (196 x^{2} + 266 x + 19 (14 x) + 19 \cdot 19) + 7 (14 x + 19) + 30$

Multiply $14$ by $19$ .

$g (14 x + 19) = - 13 (196 x^{2} + 266 x + 266 x + 19 \cdot 19) + 7 (14 x + 19) + 30$

Multiply $19$ by $19$ .

$g (14 x + 19) = - 13 (196 x^{2} + 266 x + 266 x + 361) + 7 (14 x + 19) + 30$

Add $266 x$ and $266 x$ .

$g (14 x + 19) = - 13 (196 x^{2} + 532 x + 361) + 7 (14 x + 19) + 30$

Apply the distributive property.

$g (14 x + 19) = - 13 (196 x^{2}) - 13 (532 x) - 13 \cdot 361 + 7 (14 x + 19) + 30$

Simplify.

Multiply $196$ by $- 13$ .

$g (14 x + 19) = - 2548 x^{2} - 13 (532 x) - 13 \cdot 361 + 7 (14 x + 19) + 30$

Multiply $532$ by $- 13$ .

$g (14 x + 19) = - 2548 x^{2} - 6916 x - 13 \cdot 361 + 7 (14 x + 19) + 30$

Multiply $- 13$ by $361$ .

$g (14 x + 19) = - 2548 x^{2} - 6916 x - 4693 + 7 (14 x + 19) + 30$

Apply the distributive property.

$g (14 x + 19) = - 2548 x^{2} - 6916 x - 4693 + 7 (14 x) + 7 \cdot 19 + 30$

Multiply $14$ by $7$ .

$g (14 x + 19) = - 2548 x^{2} - 6916 x - 4693 + 98 x + 7 \cdot 19 + 30$

Multiply $7$ by $19$ .

$g (14 x + 19) = - 2548 x^{2} - 6916 x - 4693 + 98 x + 133 + 30$

Step 4

Simplify by adding terms.

Add $- 6916 x$ and $98 x$ .

$g (14 x + 19) = - 2548 x^{2} - 6818 x - 4693 + 133 + 30$

Simplify by adding numbers.

Add $- 4693$ and $133$ .

$g (14 x + 19) = - 2548 x^{2} - 6818 x - 4560 + 30$

Add $- 4560$ and $30$ .

$g (14 x + 19) = - 2548 x^{2} - 6818 x - 4530$

Find g(f(x)) f(x)=14x+19 g(x)=-13x^2+7x+30

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