How do you find the inverse function of f(x) = 4x^3 + 5 when x = 3?

To find the inverse of the function f(x) = 4x³ + 5, we’ll follow a few steps:

1. Rewrite the function: Set y equal to f(x):
   
   y = 4x3 + 5
2. Swap x and y: This step is essential for finding the inverse.
   Now we rewrite the equation as:
   x = 4y3 + 5
3. Solve for y: We need to isolate y. First, subtract 5 from both sides:
   x - 5 = 4y3
4. Now, divide both sides by 4:
   (x - 5) / 4 = y3
5. Take the cube root: To solve for y, take the cube root of both sides:
   y =
oot{3}{(x - 5) / 4}
6. Write the inverse function: Thus, the inverse function is:
   
   f-1(x) =
oot{3}{(x - 5) / 4}

Now, to determine the value of the inverse function when x = 3:

1. Substitute 3 in the inverse function:
   
   f-1(3) =
oot{3}{(3 - 5) / 4}
2. Calculate the expression within the cube root:
   
   f-1(3) =
oot{3}{(-2) / 4} =
oot{3}{-0.5}
3. Since negative values can exist within cube roots, we find that:
   
   f-1(3) = -
oot{3}{0.5}

Final Answer: The inverse function is f-1(x) =
oot{3}{(x - 5) / 4}, and when x = 3, the value is f-1(3) = -
oot{3}{0.5}.