Given the function f(x)=3x^3+2, find the value of x so that f^-1(x)=4. Thank you.

To find the value of x such that f^(-1)(x) = 4, we need to find the inverse function of f(x), and then set it equal to 4.

Let's start by finding the inverse of f(x):

Step 1: Replace f(x) with y.
y = 3x^3 + 2

Step 2: Swap x and y.
x = 3y^3 + 2

Step 3: Solve the equation for y.
Subtract 2 from both sides:
x - 2 = 3y^3

Divide both sides by 3:
(x - 2) / 3 = y^3

Step 4: Take the cube root of both sides to solve for y:
y = ∛[(x - 2) / 3]

Now that we have the inverse function, f^(-1)(x), we can set it equal to 4 and solve for x:

f^(-1)(x) = 4
∛[(x - 2) / 3] = 4

Cube both sides to remove the cube root:
[(x - 2) / 3] = 4^3 = 64

Multiply both sides by 3 to eliminate the fraction:
x - 2 = 3 * 64 = 192

Add 2 to both sides to isolate x:
x = 192 + 2 = 194

Therefore, the value of x that satisfies f^(-1)(x) = 4 is x = 194.