What is the inverse of the function f(x) = x^4 + 12?

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The function given is f(x) = x4 + 12. To find the inverse of this function, we need to follow a series of steps.

  1. Replace f(x) with y:
    Start by rewriting the function as:

y = x4 + 12

  1. Switch x and y:
    To find the inverse, we exchange the roles of x and y:

x = y4 + 12

  1. Solve for y:
    Our goal now is to isolate y. Start by subtracting 12 from both sides:

x – 12 = y4

  1. Take the fourth root:
    Next, we take the fourth root of both sides to solve for y:

y = (x – 12)1/4

  1. Write the inverse function:

Now that we have y isolated, we can express the inverse function:

f-1(x) = (x – 12)1/4

In conclusion, the inverse of the function f(x) = x4 + 12 is:

f-1(x) = (x – 12)1/4

Keep in mind that the domain of the original function affects the range of the inverse function, and vice versa. Since f(x) = x4 + 12 shifts the output upwards by 12, the range of f(x) starts at 12 (i.e., y ≥ 12). Therefore, the domain of the inverse function is x ≥ 12.

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