Find the rule for the function.

Question

Let f(x) = x − 9,

g(x) = x + 9

,

and h(x) = 4x^3 − 9.

(gh)÷(g-f)

Answer 1

The objective if to find the rule or result of multiplying the function g(x) times the function h(x) and dividing the result by the result of subtracting the functions f(x) from the function f(x), or (gh)/(g-f).

Let's find the result of the first operation (gh), or g(x) * h(x).

g(x) * (h(x) = (x+9) * (4x^3-9) = 4x^4 + 36x^3 - 9x - 81

Let's find now the result of the second operation (g-f), or g(x) - f(x).

g(x) - f(x) = x + 9 - (x - 9) = x + 9 - x + 9 = 18

So, the result, or rule for the function (gh)/(g-f) will be:

(gh)/(g-f) = (4x^4 + 36x^3 - 9x - 81) / (18)

or:

(gh)/(g-f) = (2/9)x^4 + 2x^3 - (1/2)x - (9/2)