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## Determining whether a function is invertible

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# Determining if a function is invertible

CCSS.Math: ,

## Video transcript

- [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, so 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. So the function is going to, if you give it a member of the domain it's going to map from
that member of 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. So 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 is 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 fo it to
be invertible you need a, you need a function that could take go from each of these points to, they can 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, so 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, woops, the,
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 these are a, 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.