Question

let $f(x) = 3x^2 - 2x + 6$ and $g(x) = 7x - 4$. identify the rule for $g(f(x)) + g(x)$.
a. $21x^2 - 7x + 34$
b. $21x^2 - 21x + 50$
c. $147x^2 - 182x + 62$
d. $21x^3 - 26x^2 + 57x - 28$

Explanation:

Step1: Compute $g(f(x))$

Substitute $f(x)$ into $g(x)$:
$$g(f(x)) = 7(3x^2 - 2x + 6) - 4$$
$$= 21x^2 - 14x + 42 - 4$$
$$= 21x^2 - 14x + 38$$

Step2: Add $g(x)$ to $g(f(x))$

Combine the two expressions:
$$g(f(x)) + g(x) = (21x^2 - 14x + 38) + (7x - 4)$$
$$= 21x^2 + (-14x + 7x) + (38 - 4)$$
$$= 21x^2 - 7x + 34$$

Answer:

A. $21x^2 - 7x + 34$