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Inverse Functions & Relations
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Mar 26, 2025
In this lesson, we'll learn how to identify whether two functions are inverse of one another. We'll also learn how to find the inverse of a relation and a function. Chapters: 00:00 Introduction 01:02 What are inverse relations? 03:17 How to tell if two functions are inverse of one another? 09:46 How to find the inverse of a function algebraically?
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0:00
be quiet so we're doing inverse function
0:02
and relations we're done with fo gx but
0:05
we're still going to see it with inverse
0:08
functions right so I'm hoping that
0:10
you've mastered that a little bit better
0:12
than you did before all right so again
0:15
for quarter four it's a shorter quarter
0:17
in my opinion things go quick so stay on
0:20
top of things do not goof off in the
0:22
classroom no no time for jokes because
0:24
I'm actually going to give detention if
0:26
I see people still fooling around i'm
0:27
not doing that because I need to get
0:29
ahead all right so we want to talk about
0:33
inverse functions and relations so
0:36
relations are this type of stuff that
0:39
you see here i this is called a relation
0:42
right a relation because I'm creating a
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relation here a function is f of x g of
0:46
x that's a function right all right so
0:48
now two relations are inverse relations
0:51
if and only if one relation contains the
0:54
element AB and the other one has BNA
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basically
0:59
uh think of this like if I'm going to
1:04
Jack right now right i can walk up to
1:05
Jack and Jack can walk back up to me so
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that would be considered a relation
1:09
right a to B B to A so this is A this is
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B so A to Jack Jack to Mr sliming that's
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that's a relation and that's it the
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inverse cuz you can go back okay now for
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example if I have this here relation A
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is what 1 7 2
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913 and if you look at B what do you see
1:28
that's interesting inverse it's inverse
1:31
so these two relations are inverse of
1:32
one another this basic stuff right
1:35
doesn't need doesn't um require some
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kind of um a high degree of thinking so
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you're just looking at it this is 17
1:44
2913 b is 71 92 -31 so therefore these
1:50
relations are inverse of one another so
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when it says find the inverse is just
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the opposite of Yeah if I say relation A
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is 17
1:57
2913 find its inverse all you have to do
2:00
is just flip the numbers that's it so
2:02
when that happens you can conclude that
2:04
the relation the two relations are
2:06
inverse of one another all right now if
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one of them fails it's not inverse they
2:12
all have to be has to be all the way for
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example if I have let's say I give you
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another one a 0 1 uh -3 5 and then 7 and
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nine and then
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B is uh 1
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zero right 5 -3 and then 9 and three
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that will not be inverse because one of
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them is missing here right 7 9 I have 93
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here so these two relations are not
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inverse it has to be every element that
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has to satisfy this condition does that
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make sense all right so these are not
2:51
going to be called inverse because one
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of them is
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missing okay now now that we're done
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with relations we're going to talk about
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functions so one would it's the same
3:01
same principle but now with functions is
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a little different here so two function
3:05
f ofx and g of x are inverse if f of g
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ofx is equal to x and g of f ofx is
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equal to x when that happens you can
3:16
conclude that the two relations are
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inverse of one another right so now to
3:19
do that how do you prove it so I have
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two functions here i have f ofx is 3x +
3:25
9 and then g of x is 1/3 x - 3 and I
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want to show that these two functions
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are inverse of one another right so
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according to the uh the theorem f of g
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of x has to be has to be equal to x and
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g of x has also to be as also hold I
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can't speak English anymore has to be
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also equal to x so therefore if that
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happens the functions are inverse of one
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another right so let me prove that so
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I'm going to find f of g of x how would
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I do that how do I find f of g of x what
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does that mean in terms of this problem
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here what does that mean you take
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the3 x - 3 and put it into the 3x + 9 x
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excellent so you plug it in here right
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so I'm going to have 3
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* 1/3 x - 3 + 9 right let me see what
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this gives me okay so that this gives me
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what 3 * 1/3 is equal to x x 3
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*39 + 9
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this is gone right so I'm left with what
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x so you see I've proven this one I have
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to do it for this one as well it has to
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be both it can't just be one of them
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right so now I'm also going to do the
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same thing I'm going to find g of f ofx
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so what is that going to
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be now I'm going to do what I'm going to
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take this Right and I'm going to plug it
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in where
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here right so that's going to be 3 * the
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3x - 3 + 9 okay x
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oh wait that was plus 3
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isn't that the other way 3 * Oh yeah
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yeah the other way 3 * 3x + 9 no
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one * 3x3
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okay hold on
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i think
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so f(x) into that right so/3
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times 3x +
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9 - 3 right so 1/3 * 3x that is uh x 1/3
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* 9 that's 3 3 - 3 that that's gone as x
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so you see what
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happens f of g ofx is equal to x g of f
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ofx is equal to x what's the conclusion
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the two functions are what inverse very
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simple stuff right doesn't require you
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to become a rocket scientist it doesn't
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require you to be super intelligent you
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know so it's really basic stuff just
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plug it in understanding the the theorem
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and that's it but the problem lies when
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you have like uh let me give you an
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example in the book for example they
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have this function two function
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here f ofx is
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= 2x^2 -
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1 and then g of x is equal to roo
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of x + 1 / 2 right so we have these two
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function two functions and we want to
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show that these two functions are
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inverse of one another it's still the
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same principle right we're going to find
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f of g of x first right so what I'm
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going to do is I'm going to take this
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entire entity and I'm going to plug it
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in
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here okay so it's going to be two
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square of x + 1 / two the whole
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thing squared minus one right so now
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here's the thing when you when you uh
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square you take the square root of a
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radical and you you you raise to the
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power of two all you have to do is just
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get rid of the radical that's it right
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so basically what I'm saying is this
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square root of x like that square is
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equal to
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x that's it right you get rid of the
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radical
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okay so this is going to just give you 2
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* x + 1 / 2 - 1 right and what happens
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here these two are going to do what
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they're going to cancel out so you're
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left with x + 1 - 1 which is x right so
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that's f of g of x now we're going to
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have to find g of f of x as well so we
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go g of f of x so that means taking this
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entire function and plug it in for x
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here right so it's going to be square
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root of 2 x^2 - 1 right + 1 /
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2 does that make sense are we good on
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that right so that gives me that so
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2x² that'll be
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2x² over two right because 1 and 1 going
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to cancel out so what's 2x2 square over
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two is square root of what x x square
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which is X x right so therefore of X
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square is X i'm assuming that X is a
8:29
positive value that's why I can do that
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if it's done then I have to take the
8:32
absolute value but we're not worried
8:34
about that right now right we are
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assuming that x is positive so because
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this is true and this is true the two
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functions are what inverse that's it all
8:43
right this section is pretty pretty
8:45
pretty fun all right
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go ahead go very rapidly so I can jump
8:50
onto the next one here
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did you take it
9:04
all right now we're going to learn how
9:05
to find the inverse of the function
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so now
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maybe you you might have done it in
9:13
algebra one i don't know but we we're
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going to learn how to find the inverse
9:18
of a function right so let's say I have
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this function here f(x)= 2x - 1 and I
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say
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find the inverse of
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f the inverse of f i want you to find
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this
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right so the inverse function the
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notation for that right every time you
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see this fative1 of x it
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means what do you think it means
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excuse me negative no it means u index
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inverse function of f anytime you see
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that that's what that means right
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so function so if I if you ever see this
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that means I want you to find the
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inverse function of this f1 any function
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with the power of1 means that you are
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finding the inverse function that's what
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that means right so now how you find the
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inverse function of x very simple
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process again it's just a process that
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we use so what we do is this step number
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one replace
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f ofx by
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y so in this case what is that going to
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look like y = y = what 2x - 2x - 1 right
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and then the next thing is this you're
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going to solve for x step
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two
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solve for x right some people do it
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different way i like it i like it this
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way right solve for x so we're going to
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have y is 2x - one right so I'm going to
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solve for x that gives me uh 2x is equal
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to
10:56
what 1 + y y and then divided by two
11:02
right so I get x = 1 + y / 2 am I Did I
11:08
lose anybody here
11:10
huh
11:12
all right so what did I lose you
11:17
yeah
11:18
where
11:20
one
11:21
get one oh I'm sorry for here so
11:24
basically we're trying to find the
11:26
inverse function right so to find the
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inverse function there's there's some
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step that we have to take the first step
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is we need to replace f ofx by y you get
11:34
that and then here we're going to solve
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for x okay solve for x so if I'm solving
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this for x right I have y is equal to 2x
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- 1 so what's my first step add one to
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the left side of the equation add one
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right that's where the plus one came
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from you see now
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see how you should go around right and
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then from here I'm going to divide what
12:01
by two okay so I get y + 1 / 2 = x and
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now the last step again is ridiculous we
12:09
switch x and y again now we switch x and
12:11
y okay switch x and y that's just a step
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that's how we do it so now instead of y
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here we're going to have x + 1 / 2 is
12:20
equal to y you see what I did i switch x
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and y and then now you're done once you
12:25
get here you're done so therefore you
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conclude that the inverse function f1 of
12:31
x is x + 1 / 2 okay it's very simple
12:35
stuff we're just going to have to do a
12:37
couple more problems and that's going to
12:38
become more comfort you got you guys
12:40
going to get more comfortable with it
12:41
right so let me do another one here walk
12:44
you through the process and then we will
12:46
understand
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that so let's assume that I want to find
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the inverse function of f ofx is equal
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to 2x - 1 and I want to
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find
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f1x what's the first step
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replace f of x by y so I get y = 2x - 5
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then then plus on the other side to do
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what to solve for to solve for x all
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right so we get y + 5 = 2x then you
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divide the two so 5x + y 2 = x all right
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so we get y + 5 over 2 = x and then
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switch
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the x and y x and y so we get x + 5 over
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2 = y okay and now what's the inverse
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function x + 5 / 2 x + 5 / 2 so it's
13:45
really just simple algebra stuff okay
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once you understand this uh this concept
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you just have to apply it all right
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there's no rhyme no reason why they do
13:55
that it's just how they do it
13:58
okay so now uh what I want you to do is
14:01
uh I want you guys to um
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uh let's see
14:16
so what if you have this here let's
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let's do something a little bit
14:19
different so what if I have um something
14:24
with a square what is that going to look
14:26
like so let's say we have f ofx
14:30
= 2 x^2 - 10 and I want to find the
14:34
inverse
14:35
function what am I going to do
14:39
equal to y all right so we got y = 2
14:45
and then what 10 all right
14:51
mhm
14:53
square rooted right why square root
14:55
isn't because all right so we get y +
15:00
10 equals to x and then um then switch
15:04
the x and the y so we get x + 10 = y and
15:08
then and then it's f - x
15:14
= x + 10 all right see that simple
15:17
that's all we're doing here okay this
15:19
section is really So I'm going to try
15:21
and see if I give you a quiz on it so
15:22
that way you get the maximum amount of
15:24
point that you can get make it easy for
15:26
you what's the quiz not today i'm saying
15:29
like when we do it okay so I will redo
15:32
this section because you seem to like it
15:34
right all right so um he's already
15:40
Yeah so I want you guys to try this on
15:42
your own
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try this to see if these two function
15:45
might
15:55
No
16:01
actually yeah I can
16:08
no
16:10
can I borrow my
16:26
Do you own paper
16:29
all right go ahead and do this for me
16:30
what you need
16:33
are we going to do this in or can we
16:34
just do it on No this is just one
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question I'm asking i This is just
16:38
practice i want you to show that these
16:40
two functions here are inverse of one
16:43
another okay go ahead and prove prove it
16:45
to me