Which of the following graphs represent functions that have inverse functions? Why? Explain your answer clearly for credit.

Solution:

The first graph does not represent the graph of a function because this graph does not pass the horizontal line test. –This sentence is not correct. The first graph represents a graph of a function. We know this by the vertical line test.

If a horizontal line intersects a graph at two or more points, these points will not define a function when their coordinates are reversed. A function passes the horizontal line test when no two different ordered pairs have the same second component. For example draw a horizontal line with the coordinates (1,1) and (–1,1). These two ordered pairs have the same second component. The inverse coordinates would be (1,1) and (1, -1). A function provides exactly one output for each input. The input 1 is associated with two outputs, signaling that this graph inverse of this graph does not define a function.

The first graph represents a function but it is not one-to-one. Thus, it does not have an inverse function.

The second graph passes the horizontal line test and if two points were selected there ordered pairs do not have the same second components. This signifies a one-to-one function. Only one-to-one functions have inverse functions. Therefore, the second graph has an inverse function.

Solution by Lori Peddicord/Spring05