Question

How do you find the inverse of \[y = {x^3} + 5\] and is it a function?

Answer 1

Hint:
An inverse function is defined as a function, which can reverse into another function. If f and g are inverse functions, then \[f\left( x \right) = y\] if and only if \[g\left( y \right) = x\] and to find the inverse of the given function reformulate the equation with x isolated on one side to find inverse function.

Complete step by step solution:
Let us write the given function
\[y = {x^3} + 5\]
Let,
\[x = {y^3} + 5\]
To find the inverse, subtract 5 from both sides as
\[x - 5 = {y^3} + 5 - 5\]
\[x - 5 = {y^3}\]
Now take cube root on both sides of the function and transpose to get
\[\sqrt[3]{{x - 5}} = y\]
Therefore, the inverse of \[y = {x^3} + 5\]is
\[{f^{ - 1}}\left( x \right) = \sqrt[3]{{x - 5}}\]
The value of x is accompanied by only one value of y, so this relation is a function.

Additional information:
The properties of inverse functions are listed and discussed below.
Only one to one function has inverses i.e., If g is the inverse of f then f is the inverse of g. We say f and g are inverses of each other.
If f and g are inverses of each other than both are one to one function.
If f and g are inverses of each other then the domain of f is equal to the range of g and the range of f is equal to the domain of g.
If f and g are inverses of each other than their graphs are reflections of each other on the line y = x.

Note:
A function normally implies what y is if you know what x is. The inverse of a function will imply what x had to be to get that value of y. For a function that is defined to be the set of all ordered pairs (x, y), the inverse of the function is the set of all ordered pairs (y, x). The domain of the function becomes the range of the inverse of the function. The range of the function becomes the domain of the inverse of the function.