| Start | in this module we're going to learn how
to find the inverse function so if you
watch the composition of functions we
saw that the inverse function you can
you can tell us two functions are
inverses if f of g of x is the same as G
of f of X and they're both equal to X or
in other words F of G of x equals x g of
f of x equals x then f of X and G of X
are inverses inverse function so what
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| 0:41 | does that mean to be an inverse function
inverse function kind of undoes the
operation of the other function so
square root of x and x squared are
inverses cube root of x cubed is
actually slightly better example
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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| 1:13 | 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
so g of x and f of x sort of undo each other's operation that's one way to think about
inverse now y is x squared squared of X
not going to work out so well...well
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| 1:32 | 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
2 x squared 4 and g4 is the square root 2,
square four which brings me back to two
but what if i do F of negative 2, well f of
negative 2 is negative 2 squared which
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| 2:02 | is also four but G of negative 4, 4 sorry is
the square root for which is to not
negative 2 so you have to be careful
when you have functions and their
inverses what happens is there are
functions that are their numbers that
are the domain of f of X that will not
be in the range of g of x so functions and
their inverses the domain of the one
function is the range of the other and
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| 2:34 | the other way around so the input to the
first function because the output of the
second function and vice versa
another way to think about inverses
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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| 3:00 | negative 1 notation will go through the point B comma A so notice you put a you get out b
here if you plug in b get back to a
practically that has the effect of you take
the line y equals x and they are
reflections over the line y equals x
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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| 3:30 | do a straight line will be perpendicular
to y equals x distance will equal this distance
a little mathy but that's what's going
on with an inverse so the reflections
over the line
so one function looks like this other
function would look like that you can
imagine if you fold it on the line that
they go on top of each other
ok so how do we find the inverse
function well it's not always easy to do
but if you have a simple function say you
have oh you want to find the inverse function of that
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| 4:09 | that this simple way to do is with an
algorithm first replace f of X with y so
replace
okay you have two steps you switch x and y
and then you isolate y so here
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| 4:45 | if you want to switch x and y so this
would have been a written as to 2X plus 3
great so step one switch X&Y so it
becomes X is 2 y plus 3 and now you want
isolate y so subtract 3 from both
sides and divide by two
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| 5:07 | that's y and thats the inverse function so
the inverse of f of x equals 2x plus 3 is
F inverse of X is X minus 3 over 2
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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| 5:38 | plug it in there two times X minus 3
over 2 plus three and you do the cancelations of x
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
plus 3 minus 3 over 2 which is x so
remember think that works inverses of
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| 6:05 | each other so let's do it again with a slightly messier function
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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| 6:32 | replace f of X with Y so Y is 5x cubed plus 4
over 7 our first steps which x and y
x is 5 y cubed plus 4/7 and now we're going
to isolate y so cross multiply subtract
4 from both sides / 5 and take the cubed
route and that is the inverse function
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| 7:18 | remember what I said the one function
undoes the others if you think about it
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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| 7:32 | add 4 the last you divide by 7 so now here you
do the other working
divided by seven so the first thing you
do is you multiply by 7 then you add 4
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
thing you do here is cubed root it and that's another
way to think about inverses so we're
do inversus show up, so where i showed you
square root cube root fourth root those
are inverses of squared cubed
fourth sine cosine and tangent all the
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| 8:04 | trig functions have inverses inverse
sine inverse cosine inverse tan and
logarithms and Exponential's are inverse
are the main places you're going to see
inverse functions in precalculus and
calculus
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