Which of the following graphs shows a function that has an inverse? (see the image for graphs) a. none b. I only c. II only d. I and II e. I, II, and III
A function is a one-one relationship between two variables `x` and `y` ie each value of `x` is mapped to one value of `y` only.
Two values of `x` may map to the same value of `y` and the relation is still a function. However, if` ` a value of `x` maps to more than one value of `y` the relation is not a function (it is a one-many relationship).
Only functions where each and every value of `y` corresponds to only one value of `x` have inverses. The function `x -> y` can only have an inverse if `y -> x` is also a function (is one-one).
So, let's look at each graph in turn
I) Each value of `x` corresponds to one value of `y` and vice versa. We can see this is true because the graph is increasing with `x`. The relation is a function and the function has an inverse function. The inverse function is the reflection of the function in the line y = x. Try doing this and you will see that the inverse function takes the shape of an S-curve.
II) This is a parabola. Each value of `x` corresponds to one value of `y` so the relation is a function. However, more than one value of `x` maps to the same value of `y` as the graph is symmetric about its vertex. Therefore `y->x` is one-many and hence is not a function. Reflecting the graph in the line y = x would result in a graph much like III) which is not a function.
III) This is a parabola reflected in the line y = x and is a one-many relation and so not a function. Some values of `x` correspond to more than one value of `y`.
Answer is b) Only graph I has an inverse