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+4x-2=14
g(x)= x^3+4x-16=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+4x-2
x= y^3+4y-2
x-2=y^3+4y
But I don't know how to solve for y.
Any help would be appreciated. Thank you.
To find g(14), we need to find the value of x for which h(x) equals 14. However, since you're having trouble solving the equation h(x) = 14 directly, we'll take a different approach.
Since g(x) represents the inverse of h(x), if (a, f(a)) is on h(x), then (f(a), a) will be on g(x). In other words, if we can find x such that h(x) = 14, then g(14) will equal x.
Let's set up the equation h(x) = 14:
x^3 + 4x - 2 = 14
To solve this equation, we want to isolate the x term. Start by subtracting 14 from both sides:
x^3 + 4x - 16 = 0
Now, finding the exact value of x that satisfies this equation can be challenging since it's a cubic equation. One option is to use numerical methods or a graphing calculator to estimate the solution.
Alternatively, you can use a numerical root-finding algorithm like the Newton-Raphson method to approximate the root. However, this method can be complex, especially if you haven't covered it in your class yet.
If finding an exact solution is not required, you might consider using an online cubic equation solver or equation-solving software to find an approximate solution for x. Once you have that value of x, g(14) will be equal to that approximate value.
Remember, it's not always necessary to find the explicit equation for g(x) to solve problems involving inverse functions. In this case, finding the exact value of x may be challenging, so an approximate solution may be sufficient.
Yoiu understand correctly what you need to do, but you should write:
"g(x)= x^3+4x-16=0"
Instead, you can say that if x = g(14), then x satisfies the equation:
x^3+4x-16 = 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.