inverse function problem

Question

Give the equation of a function whose inverse is also a function. Then, draw a sketch of a non-function whose inverse is also a non-function 

Answer 1

f(x) = x

A circle.

Answer 2

Any of the equations from your previous question (function equals its inverse) would also answer the first part of your question.  The key to a graph being a function is the vertical line test.  If it also passes the horizontal line test, then it's inverse is also a function.  So, f(x) = x is a function, and its inverse is also a function.  You can easily verify this graphically.  f(x) = x² would be an example of a function whose inverse is a non-function, since the graph of a parabola passes the vertical line test but fails the horizontal line test.  So, the second part of your question looks for a sketch of a curve that fails both the vertical and horizontal line tests.  A circle would be a simple example, with equation x² + y² = 1.  If you don't need the equation, you're welcome to sketch any wiggly curve you want that doubles back enough to fail both the vertical and horizontal line tests.

 

If vertical/horizontal line tests still confuse you, you really ought to set up an online lesson with somebody so we can draw these examples on a whiteboard for you.

Answer 3

Give the equation of a function whose inverse is also a function.  y = x+1 is a function and because it's strictly increasing it passes the HORIZONTAL LINE TEST and therefore it has an inverse.

 

sketch a non-function whose inverse is also a non-function.

x² +y² = 1, the unit circle, is not a function.

solving for y after interchanging x & y (to attempt to find an inverse) yields a non-function, too--the same circle.