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Definition of One-to-One Functions - Problem 1

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Determining if a relationship is a function and if a relationship is 1 to 1. So I have a chart right here which has some data points and we’ll call this first column x and this second column y. What we are concerned with if it’s a function for every x there’s only one y or if it’s 1 to 1 for every x there’s y of for every y there’s just one x.

So looking at this, 1 goes to 5 2 goes to 6. So those are both 1 to 1 relationship for every x there’s only one y for every y there’s only one x. What we are concerned with here is 3 and 4 both going to 7. So the function part for every x there’s only one y.

If x is 3 y is 7, if x is 4 y is 7. So yeah for every x there’s only one y which means this is going to be a function. Second part is it one to one for every y is there only one x? And what we see here is if y is 7 there’s actually two x values so that tells us this is not actually 1 to 1.

Another example, here we have a different relationship again we are relating x and y for here this is just a straight or cross relationship. There’s x there’s a y and then there’s for every one x there’s one y and vice versa for every y there’s one x. So first part tells us that’s a function one x is one y. The second part for every y there’s also only one x tells us this is 1 to 1 as well. So the function is also one to one.

Our third example, our last example I want to see if this is a function. First part of the function is for every x is there only one y? Looking right here at number 1, if x is 1 we have y 5 x is 1 we also have y 6. So this tells us it’s not actually a function because for one x, number 1 there’s actually two different y’s. So this is not a function if it’s not a function it also cannot be 1 to 1 because part of being 1 to 1 is at first clause which means it also has to be a function. So if you have no for function it can’t be 1 to 1.

So just some examples of looking at some data making sure for every x there’s one y for a function and also for every y there’s one x just to make sure it's 1 to 1.

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