Calculus
posted by Kelly .
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.
Respond to this Question
Similar Questions

inverse
find f^1 (x). (this is asking me to find the inverse) f(x) = (x2)^2, x <= 2 how do I solve this problem? 
inverse
If f(x)=cosx + 3 how do I find f inverse(1)? 
Calculus
F(x) = 1x and g(x) = 1/x These functions have the property that f = f^1(inverse) and g = g ^1. That is, the inverse of f is equal to itself and the inverse of g is also equal to itself. Take the composition of each function with … 
pre calculus
Find the inverse of the function below. Graph the function below and the inverse. Determine the domain, range and asymptotes of the function below and the inverse function. Please show all your work. f(x) = 2e^x + 5 Just looking at … 
precalculus
the function f(x)=4/x7 is one to one a. find the inverse of f b. graph f, f^1, and y=x on the same set of axes find the inverse of f f^1(x)= 
Algebra
T o convert from X degrees centigrade to y degrees fahrenheit the function is f(x) 9/5x +32 find the inverse function f of negative 1. what does the inverse function represent? 
PreCalculus
g(x)= x^2+7 Find the inverse of g(x) and state the domain and range for the inverse of g(x) using interval notation 
Calculus
If g(X) = 2+ln(x5), find the inverse function, (g^(1))(x) Solution: y = 2+ ln(x5) e^y = 2 + x  5 x = e^y + 3 y = e^x + 3 g^(1) = y = e^x + 3 Is that right? 
Algebra2
Does the relation in the table represent direct variation, inverse variation, or neither? 
Math
Does the relation in the table represent direct variation, inverse variation, or neither?