Answer:
f^-1(x) = 2(x-2)^2 + 1
f^-1(x) = (x+3)^3
Explanation:
The inverse of a function is found by switching the roles of the dependent and independent variables. In other words, we can find the inverse of a function by interchanging the x and y values in the equation. So, the inverse of the function f(x) = 2(x-2)^2 + 1 can be written as f^-1(x) = 2(x-2)^2 + 1.
1. To find the inverse function explicitly, we can use the following steps:
2. Replace f(x) with y in the original function to obtain y = 2(x-2)^2 + 1.
3. Solve for x in terms of y to obtain x = +/- sqrt((y-1)/2) + 2.
The inverse function f^-1(x) can then be written as f^-1(x) = +/- sqrt((x-1)/2) + 2.
Note that this inverse function is not a one-to-one function, because for some values of x (namely, x < 1), the function has two possible values of y (i.e. it has two solutions). Therefore, the inverse function is not a function in the strict mathematical sense, but it is still a useful concept in certain contexts.
To find the inverse of a function, we switch the roles of the dependent and independent variables, so the inverse function can be written as x = ^3√y-3.
1. To find the inverse function explicitly, we can use the following steps:
2. Replace y with x in the original function to obtain x = ^3√y-3.
3. Solve for y in terms of x to obtain y = (x+3)^3.
The inverse function can then be written as f^-1(x) = (x+3)^3.
Note that this inverse function is a one-to-one function, meaning that for every value of x, there is only one corresponding value of y. Therefore, the inverse function is a function in the strict mathematical sense.