Calculus
posted by Kelly on .
Let h(x)= x^3 + 4x 2. Let g(x) represent the inverse of h(x).
Find g(14).
I know that if (a, f(a)) is on h(x), then (f(a), a) will be on g(x). I also know that if f^1(a)=b if f(b)=a. I thought that if I could find the x value(s) at which h(x)=14, I would be able to find g(14).
g(x)= x^3+4x2=14
g(x)= x^3+4x16=0
I don't know where to go from here. I don't know how to solve the cubic to find x.
My teacher said it was not necessary to find the equation for g(x) to solve the problem. I tried finding it at first, but ran into problems there as well:
if y= x^3+4x2
x= y^3+4y2
x2=y^3+4y
But I don't know how to solve for y.
Any help would be appreciated. Thank you.

Yoiu understand correctly what you need to do, but you should write:
"g(x)= x^3+4x16=0"
Instead, you can say that if x = g(14), then x satisfies the equation:
x^3+4x16 = 0
Use the rational roots theorem. Since 16 has many divisors, you can try to shift x, e.g. put x = t + 1.
For the Rational Rpoots theorem, you only need to know the coefficient of t^3 and the constant term. The coefficient of t^3 is 1 and the coefficient of the constant term is the value of the polynomial at t = 0, which corresponds to x = 1, so this is 11.
So, the only possible roots are
t = ±1 and t= ±11.
Add 1 to find the possible roots for the polynomial as a function of x:
x = 0, 2, 10, 12
If we apply the rational roots theorem to the original polynomial directly, then we find that the possible roots are powers of two up to a sign till 16. But we also know that x must be among the above list, so x = 2 is the only possible rational root.
If you try out x = 2, you see that it is indeed a zero.