f(x)= radical sign (6x-30) for the domain [5,∞)

Question

Find f^-1(x) where f−1 is the inverse of f

Also state the domain of f^-1 in interval notation

f^-1(x) = _ for the domain _

Answer 1

For a function and its inverse function, the domains and ranges of each function switch places. Therefore:

Domain of a function is the same as the Range of the inverse function

Range of a function is the same as the Domain of the inverse function.

To find the inverse of a function, replace the function name with y, switch all x and y-variables, and resolve the equation for y:

f(x) = √(6x-30) --> y = √(6x-30)

x = √(6y-30)

x² = 6y-30

x² + 30 = 6y

y = (x²+30)/5 --> Therefore, f^-1(x) = (x²+30)/5

The range of the original function, f(x), is the domain of the inverse function. For the original function, the radical will only produce y-values greater than/equal to 0, therefore the range of this value/domain of the inverse function are all values greater than/equal to 0. Otherwise written as [0, inf)