Question

For the function f(x)=5(x^7/7)^1/5, find f^-1(x)

Answer 1

To find the inverse function f^{-1}(x) for the function f(x) = 5(x^7/7)^{1/5}, we swap x and y, manipulate the equation to solve for y, and then express y independently in terms of x, which results in f^{-1}(x) = (7x^5/5^5)^{1/7}.

Step-by-step explanation:

Given the function f(x) = 5(x7/7)1/5, we want to find the inverse function, denoted as f-1(x). To find the inverse, we need to swap the roles of x and y and solve for y after substituting f(x) with y. Here are the steps:

1. Write the function as y = 5(x7/7)1/5.
2. Replace y with x and vice versa, resulting in x = 5(y7/7)1/5.
3. Raise both sides to the power of 5 to get rid of the exponent on the right hand side: x5 = 55(y7/7).
4. Divide both sides by 55 to isolate y7: x5/55 = y7/7.
5. Multiply both sides by 7: 7x5/55 = y7.
6. Take the 7th root on both sides to solve for y: y = (7x5/55)1/7.
7. Therefore, the inverse function is f-1(x) = (7x5/55)1/7.