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Hi everyone today we're going to talk about how to determine whether a function is one to one or in other words whether it passes the horizontal line test to complete this problem we'll look at several methods we can use to determine whether the function is one to one let's take a look in this particular problem we've been asked to determine whether these two functions are one to one keep in mind that's that the same thing as asking whether these functions pass the horizontal line test the function could be called one to one and passes the horizontal line test if there's only one x output value for every y input value so its really just the opposite of the vertical line test to solve a problem like this we have to show that every x input value gives a different y output value and if we have two different x inputs that give the same y output then we know that the function is not one to one so for example for the function like f of x where we have f of x equals x squared minus two x if we factor this factor out an x and get x times x minus two if x is equal to either zero or two the result will be zero so what we're saying that if we plug in zero for x we'll get zero times zero minus two which will give us zero if we're plugging two for x we'll get two times two minus two which will also give us zero so what we said is that we can plug in either zero or two for x and both of those answers both of those values give us zero for an answer so because we've plugged in two different x values and gotten back the same y value we know that this function is not one to one if we look at our second function g of f its really easy to see that no matter what value we plug in for x we'll always get a unique answer back for y because if we plug in negative one zero one two three four any value here for x this function will return the different value for y, so we know right away that this function is one to one another way to evaluate this is to assume that g of a or you could do this with our function f of x too in which case you'll have f o a bit the g of an is equal to g of b if g of an equals g of b then we know that a has to equal b so if we can show that by plugging in a to g and b to g and then getting a to equal b we can show that the function is one to one so what we say here is if we plug in a for x and b for x and set those equal to each other we'll get one over an equals one over b, so you're just plugging in a to your function plugging in b to your function and then setting those two things equal to one another if you can do that and then solve this equation to get down to the exact equation an equals b then you know your function is one to one and of course we can do that because here if we multiply both sides by a and then multiple both sides by b we'll get b equals a, and so we know that our function is one to one, so that's another trick you can use to figure out whether the function is one to one you can also always use the graph of the function...
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