Not Every Function Has a Inverse Function

Many functions, such as y = x², do not have inverse functions: In Figure 2.4.3, we see that x is not a function of y because at y = 1, x has the two values x = 1 and x = -1.

Often one can tell whether a function f has an inverse by looking at its graph. If there is a horizontal line y = c which cuts the graph at more than one point, the function f has no inverse. (See Figure 2.4.3.) If no horizontal line cuts the graph at more than one point, then f has an inverse function g. Using this rule, we can see in Figure 2.4.4 that the functions y = |x| and y =do not have inverses.