hi everyone today we're going to talk about how to determine whether or not a function
is one to one or in other words whether or not it passes the horizontal line test to
complete this problem we'll look at several methods we can use to determine whether or
not 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 thats that the
same thing as asking whether or not these functions pass the horizontal line test the
function could be called one to one and passes the horizontal line test if theres 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 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 but the g of a is equal
to g of b if g of a 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 a equals one over b so your 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 a equals b then you know your function is one to one and of
course we can do that because here if we multiple 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 or not the function is one to one you can also
always use the graph of the function if you have it or if you know what it would look
like to determine whether or not the function is one to one obviously this function here
with f of x we have an x squared in the function and we know that functions with an x squared
are going to have a parabola like this in the graph and we know that obviously a parabola
would not pass the horizontal line test because we can draw a horizontal line that will cross
the graph at more than one place so it crosses the graph at two places so we know that it
wouldn't pass the horizontal line test and therefore wouldn't be one to one so that's
it that's a couple of ways to determine whether or not a function is one to one or in other
words whether or not it passes the horizontal line test so I hope you found that video helpful
if you did like this video down below and subscribe to notified of future videos