WEBVTT
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Language: en
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in this module we're going to learn how
to find the inverse function so if you
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watch the composition of functions we
saw that the inverse function you can
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you can tell us two functions are
inverses if f of g of x is the same as G
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of f of X and they're both equal to X or
in other words F of G of x equals x g of
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f of x equals x then f of X and G of X
are inverses inverse function so what
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does that mean to be an inverse function
inverse function kind of undoes the
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operation of the other function so
square root of x and x squared are
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inverses cube root of x cubed is
actually slightly better example
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so if f of X is X cubed, f of 2 is 2 cubed is 8 and then g of x the cube root of x and g 8 is the cubed root
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eight which is two notice first you plug
in two and you get 8 and then you plug in 8 and get back to two
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so g of x and f of x sort of undo each other's operation that's one way to think about
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inverse now y is x squared squared of X
not going to work out so well...well
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let's see let's say that f of x is x squared
g of x the square root of x if i plug in
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2 x squared 4 and g4 is the square root 2,
square four which brings me back to two
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but what if i do F of negative 2, well f of
negative 2 is negative 2 squared which
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is also four but G of negative 4, 4 sorry is
the square root for which is to not
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negative 2 so you have to be careful
when you have functions and their
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inverses what happens is there are
functions that are their numbers that
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are the domain of f of X that will not
be in the range of g of x so functions and
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their inverses the domain of the one
function is the range of the other and
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the other way around so the input to the
first function because the output of the
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second function and vice versa
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another way to think about inverses
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is if f of X goes through the point a comma B and the F inverse of X which we write this
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negative 1 notation will go through the point B comma A so notice you put a you get out b
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here if you plug in b get back to a
practically that has the effect of you take
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the line y equals x and they are
reflections over the line y equals x
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so if you have the point a comma B will be here and the point b common a be about there if you
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do a straight line will be perpendicular
to y equals x distance will equal this distance
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a little mathy but that's what's going
on with an inverse so the reflections
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over the line
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so one function looks like this other
function would look like that you can
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imagine if you fold it on the line that
they go on top of each other
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ok so how do we find the inverse
function well it's not always easy to do
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but if you have a simple function say you
have oh you want to find the inverse function of that
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that this simple way to do is with an
algorithm first replace f of X with y so
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replace
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okay you have two steps you switch x and y
and then you isolate y so here
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if you want to switch x and y so this
would have been a written as to 2X plus 3
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great so step one switch X&Y so it
becomes X is 2 y plus 3 and now you want
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isolate y so subtract 3 from both
sides and divide by two
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that's y and thats the inverse function so
the inverse of f of x equals 2x plus 3 is
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F inverse of X is X minus 3 over 2
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let's verify that so f love F inverse of
X would be you take X minus 3 over 2 and
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plug it in there two times X minus 3
over 2 plus three and you do the cancelations of x
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and you get x and then if you do f inverse of f of X now you'll take FX and put it in there so you'll get 2x
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plus 3 minus 3 over 2 which is x so
remember think that works inverses of
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each other so let's do it again with a slightly messier function
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so let's say f of x is 5x cubed plus 4
over 7 that looks very bad but it's not so
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replace f of X with Y so Y is 5x cubed plus 4
over 7 our first steps which x and y
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x is 5 y cubed plus 4/7 and now we're going
to isolate y so cross multiply subtract
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4 from both sides / 5 and take the cubed
route and that is the inverse function
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remember what I said the one function
undoes the others if you think about it
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this function you plug in the number and what
you doing first you cube it then you times 5 then you
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add 4 the last you divide by 7 so now here you
do the other working
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divided by seven so the first thing you
do is you multiply by 7 then you add 4
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i subtract 4 nd multiplied by 5 and now you divide by 5 and then the last thing you did was cubed it and last
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thing you do here is cubed root it and that's another
way to think about inverses so we're
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do inversus show up, so where i showed you
square root cube root fourth root those
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are inverses of squared cubed
fourth sine cosine and tangent all the
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trig functions have inverses inverse
sine inverse cosine inverse tan and
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logarithms and Exponential's are inverse
are the main places you're going to see
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inverse functions in precalculus and
calculus