Determining if a function is invertible | Mathematics III | High School Math | Khan Academy

• [Voiceover] "F is a finite function whose domain is the letters a to e. The following table lists the output for each input in f's domain."

So if x is equal to a, then if we input a into our function, then we output -6. f of a is -6. We input b, we get three; we input c, we get -6; we input d, we get two; we input e, we get -6.

"Build the mapping diagram for f by dragging the endpoints of the segments in the graph below so that they pair each domain element with its correct range element. Then, determine if f is invertible."

Alright, so let's see what's going on over here. Let me scroll down a little bit more. So in this purple oval, this is representing the domain of our function f, and this is the range. The function is going to, if you give it a member of the domain, it's going to map from that member of the domain to a member of the range.

So, for example, you input a into the function; it goes to -6. So a goes to -6, so I drag that right over there. b goes to three; c goes to -6. So it's already interesting that we have multiple values that point to -6. This is okay for f to be a function, but we'll see it might make it a little bit tricky for f to be invertible.

So let's see, d points to two, or maps to two. So you input d into our function, you're going to output two, and then finally, e maps to -6 as well. e maps to -6 as well.

So, that's a visualization of how this function f maps from a through e to members of the range, but also ask ourselves, "Is this function invertible?" And I already hinted at it a little bit.

Well, in order for it to be invertible, you need a function that could take each of these points to do the inverse mapping. But it has to be a function. So, if you input three into this inverse function, it should give you b. If you input two into this inverse function, it should output d. If you input -6 into this inverse function, well, this hypothetical inverse function: what should it do?

Well, you can't have a function that, if you input one, if you input a number, it could have three possible values: a, c, or e. You can only map to one value. So there isn't, you actually can't set up an inverse function that does this because it wouldn't be a function. You can't go from input -6 into that inverse function and get three different values. So this is not invertible.

Let's do another example. So here, this is the same drill. We have our members of our domain, members of our range. We can build our mapping diagram. a maps to -36; b maps to nine; c maps to -4; d maps to 49; and then finally, e maps to 25. e maps to 25.

Now, is this function invertible? Well, let's think about it. The inverse—oops, was it d maps to 49? So, let's think about what the inverse, this hypothetical inverse function, would have to do. It would have to take each of these members of the range and do the inverse mapping.

So if you input 49 into our inverse function, it should give you d. Input 25, it should give you e. Input nine, it gives you b. You input -4, it inputs c. You input -36, it gives you a. So you could easily construct an inverse function here.

So this is very much, this is very much invertible. One way to think about it is this is a one-to-one mapping. Each of the members of the domain correspond to a unique member of the range. You don't have two members of the domain pointing to the same member of the range.

Anyway, hopefully, you found that interesting.