Part 4/6:
The curve of ( y = x^3 + 2 ) rises steeply through the coordinate plane, while the curve of its inverse reflects around the line ( y = x ).
If you plot a point ( (a, b) ) on the graph of ( f(x) ), its corresponding inverse point will be ( (b, a) ).
This reflective property is a key characteristic of inverse functions, demonstrating symmetry in their graphs.
Domain and Range of Inverse Functions
When discussing inverse functions, it is necessary to consider how the domains and ranges relate. For a function ( f(x) ), the domain of ( f^{-1}(x) ) is the range of ( f(x) ) and vice versa. This swapping is essential because the input and output of the inverse function are directly exchanged.