This video shows the various methods that make the concept
crystal clear. To make the concept of inverse function easy for
the learners.
crystal clear. To make the concept of inverse function easy for
the learners.
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📚
LearningTranscript
00:00Today, I am going to discuss the topic Inverse Function.
00:09This topic is by RBI Educational Academy.
00:13Here you will see the concept in an oral way.
00:18So I request you to watch the video from beginning till end.
00:25Now see here students, what do you mean by inverse function?
00:32The inverse of a function is it reverse that is it undoes the function f at x.
00:41How it undoes?
00:42I will show you.
00:43The inverse of the function f at x is written as f inverse at x.
00:51Suppose take the example f at x, f at x is x plus 2 in one way it can be represent like
00:58this or here you see f such that function f such that x is mapped onto x plus 2.
01:10Suppose here I will take one example, here I will draw domain, encode domain.
01:18So here I will take the numbers domain 1, 2, 3, minus 1, minus 2 and f at x is a function
01:38is x plus 2.
01:41So f at x is x plus 2.
01:46So in place of x if I write f at 1, 1 plus 2 that is nothing but 3.
01:55So 1 is mapped to 3.
01:57If I take 2, f at 2 is 2 plus 2 is 4.
02:05So 2 is mapped to 4.
02:08If I take 3, 3 plus 2 is 5 and if I take minus 1.
02:31Now here f at x what I have written x plus 2.
02:47So we took the numbers 1, 2, 3 and f at x is x plus 2, 1 is mapped to 1, 1 is mapped
03:00to 5.
03:01So if I write 1, 1 is attached to 3.
03:03If I take 3, 2 plus 2 I have written 4, 3 plus 2 5.
03:08If I extend the number minus 1 and minus 2 here, minus 1, minus 1 plus 2, 2 minus 1 is
03:181 with plus sign because we take the sign of a bigger number always.
03:37So sorry for the disturbance boys, 1, 2, 3, minus 1, minus 2.
03:47So 2 plus 1, 3, 2 plus 2, 4, 3 plus 2, 5 then here I want to show minus 1 plus 2 we get
03:541.
03:55If I write minus 2 plus 2 it is 0.
03:59So minus 1 attached to 1 and 2 is attached to 0.
04:02So this was a function f at x.
04:06These variable we call it as x, 1, 2, 3 we replace x 1 time by 1, 1 time by 2, then 3,
04:14minus 1, minus 2 when we write 1, 1 plus 2 we get 3.
04:19So 1 is attached to 3, 2 plus 2 is 4.
04:23So this number 3, 4, 5, 1, 0 these are the images.
04:29Image of 1 is 3, image of 2 is 4, image of 3 is 5.
04:35So these images we represent by letter y or we call these images as f of x.
04:44It is same as f at x but f at x is a function, it is not the same.
04:48f of x is a real number, generally we say f of x this to function and to image also
04:55we say f of x but f of x is a real number and f at x is a function, please remember.
05:03Now when we talk about the inverse function, inverse function will be from co-domain to
05:11domain.
05:13So when we talk about this.
05:28So here when we talk about the inverse function, here the image of 1 is 3.
05:36So f inverse means 3 will be attached to 1, we see f inverse of y, we call this as x and
05:44this as y.
05:45So f inverse, I mean to say f inverse of y that will be nothing but it will be equal
05:51to x, this is the main thing which I want to say you.
05:54So boys here, I am going to show you how we will find f inverse of x.
06:20So f at x, I will write here is equal to x plus 2 in place of f of x, I will take y
06:30is equal to x plus 2 because x, its image is y or you can say this as f of x.
06:41So here I replace f of x by y, y is equal to x plus 2 and plus 2 when we send this side,
06:50it become minus 2, either you can subtract 2 from both side, y minus 2 is equal to x
06:58plus 2 minus 2, we will get cancel, you will get x or transposition, we move shift plus
07:052 from right to left, so the plus sign changes to minus sign, y minus 2 is equal to x.
07:11Now when we define inverse function, f inverse of y will be x, so x is equal to y minus 2,
07:24x is the subject, so I write on left hand side.
07:28Now see here, f inverse of y, if I write here f inverse of y, that will be, when I
07:37substitute y in f inverse function, I will get x, so in place of x, I can write f inverse
07:43of y is equal to y minus 2 and if I replace y by x, f inverse of x is equal to x minus
07:542, in place of y I have written x, so f at x is a function, f at x is equal to x plus
08:022 is the given function, its inverse is f inverse at x is x minus 2.
08:08Now boys, here I want to say you one more thing, you can remember in another way also
08:17that I am going to say you now, f at x, f of x, the image we call it as y, I can apply
08:30f inverse on both side, f inverse of f of x is equal to f inverse of y, on right hand
08:41side I apply an inverse function, on left hand side also I apply an inverse function,
08:47f inverse means 1 over f, f of x is equal to f inverse of y.
08:57Now boys, see here f is in the numerator, f is in the denominator, okay, you cannot
09:05cancel function, so it is better we say remove this both from the numerator and denominator,
09:11so you can say x is nothing but equal to f inverse of y, so in place of x, you can substitute
09:20f inverse of y, x is nothing but f inverse of y or you think that you substitute y in
09:28inverse function, f inverse of y, we get x, so in place of x, f inverse of y, the last
09:34step y we replace by x, that is x minus 2.
09:38Now there is one more way, okay, there is one more way to find the inverse of a function,
09:53so what is that way in which we find f, to find the inverse of a function, how we can
10:05find the inverse of a function, okay, the first step we write f at x is equal to x plus
10:112, so the first step is rewrite the function replacing f at x with phi, so I will rewrite
10:25now in place of f of x, I will write x, y is equal to x plus 2.
10:32In place of f of x, this f of x, I replace f of x with phi, see I replaced it, now the
10:41next step interchange x and y, here I mean to swap, in place of y, I will write x and
10:48in place of x, I will write y, so now it will be x is equal to y plus 2, so I swap, in place
11:00of y, I have written x, in place of x, I have written y.
11:05Now what is the third step, rearrange the equation to make y the subject, now I can
11:11subtract minus 2 from both side or send plus 2 from right to left, so it become x minus
11:202 is equal to y, okay, now you know it, I have shown you just now, here x, here y, when
11:32we substitute y in inverse function, f inverse of y is nothing but x, so in place of y, last
11:42step just write f inverse of x, f inverse of x when we apply inverse function, you get
11:52x minus 2, so this was our second method.
12:03Now I will discuss with you the third method, before graph I should have discussed, okay,
12:27but I will discuss, nothing will happen, here in the graph I will discuss, then I will show
12:32you the graph, so step a, I will write a flow diagram for f, a flow diagram for f, how x,
12:47let us say add 2, add 2 in flow diagram, so I get f at x is x plus 2, now I will write
13:02a flow diagram for f inverse, flow diagram for f inverse, here it is very easy, simple,
13:17see here, I will start from right side, okay, in flow diagram just write the inverse, add,
13:26I will write subtract, inverse of addition, subtraction, so subtract 2, we get x minus 2,
13:37so f at x is x plus 2 and f inverse at x is x minus 2, it is so simple.
13:46So, now I will discuss the flow diagram with you, just now I have discussed flow diagram with you,
13:54now I will show you the graph of f at x and f inverse at x, so 1, when I keep 1 plus 2, 3,
14:08because the function is f at x is x plus 2, this is the function, now if I keep 2, I get 3,
14:22if I keep 3, I get 4, if I write minus 1, minus 1 plus 2 is 1, if I write minus 2, minus 2 plus 2
14:32is 0, 1 max 2, 2 max 3, if I write 2, 2 max 4, minus 1 max 1, minus 2 max 0, this is the function,
14:46from domain to coordinate. Now, if I draw the graph, see the graph, so if I take 0, 0 plus 2,
15:01I will get 2, so 0 to 2, it will be here, if I write 1, 1 plus 2, it is 3, so 1 to 3 is here my
15:09boy, if I take 2, 2 plus 2, 4, so 2 to 4, it is here, if I write minus 1, minus 1 plus 2 is 1,
15:19minus 1, 2, so it is here, if I take minus 2, minus 2 plus 2 is 0, so this represent f at x.
15:30Now, let us see this, this represent what? So, let us see what it represent, it represent inverse
15:45function that is f inverse that is reverse, reverse of it, f inverse of x, so what happen,
15:53f inverse of x, just now I have done x minus 2, how it will be, if I write 0, I will get 0 minus
16:052 that is minus 2, 0 minus 2 and if you draw a line, this line is y is equal to x, you can see
16:17the symmetry, if I take 0, 0 to plus 2, in the inverse, if I write 0, 0 to minus 2, now if I
16:26write 1, 1 plus 2, 3, 1 to 3, if I write 1 here, 1 minus 2, I get minus 1 boys, so here if I write
16:411, I get minus 1 here, you can see here minus 1 and if I write 2, 2 minus 2 will be 0, you see
16:50here, I will show you, f inverse at x is equal to x minus 2, if I write 2, 2 minus 2 is 0, so x2,
17:07when x is 2, in inverse function, if I write f inverse at 2, 2 minus 2, 0, so when x is 2, y is
17:140 here, if I write 3, 3 minus 2 is 1, 3, 2, 1 here, so you see here, when I substitute 2 here,
17:251 here, 1 plus 2, it is 3, okay, when I substitute 1, I get 3 but when I substitute 1 here, I get
17:33minus 1, so see, see the symmetry, see the symmetry here, see the symmetry, okay.
17:42If I write 2, 2 plus 2, 4, I get 4, if I write in this function f inverse 2, 2 minus 2, 0, so y is
17:52equal to x, this line gives the symmetry between f at x and f inverse at x. Now, I will give you
18:05another example, if I take g at x is equal to 2x plus 4 divided by 3, okay, I have to find the
18:23inverse function, so first step in place of g at x, I write y is equal to 2x plus 4 by 3. Now,
18:35there are two ways, either multiply by 3 on both side or you can multiply 3 with y,
18:42it is 3y is equal to 2x plus 4, the division changes to multiplication when we send from
18:51right to left, so 3 multiplied with y and see here, we send plus 4 this side from right to
19:01left, so 3y minus 4 is equal to 2x. Next step, we want to make x as a subject, so I will write 2x
19:14on left hand side, 2x is equal to 3y minus 4, then x divide by 2 on both the side or send 2 on
19:29right hand side, 2 and 2 cancel, so x is equal to 3y minus 4 divided by 2. Now, see here in place
19:45of x, I can write F inverse of y, F inverse of y is equal to 3y minus 4 by 2. Now, in place of y,
20:02I replace x, last step, F inverse of x is equal to 3x minus 4 divided by 2. Now, see here,
20:15there is easy way also, so to do this all, I will show you both the ways, I will show you the
20:30swapping way and I will show you the flow diagram way, so 2x plus 4 by 3. Now, g of x is y, I think
20:58I have taken, it is g, I should not write x, I should use g only. Now, see here, g at x is y,
21:08y is equal to 2x plus 4 by 3, the first step, then in place of c at x, I have written y,
21:20then I will swap, the next step is swap. So, in place of y, I will write x and in place of x,
21:30I will write y, x is equal to 2y plus 4 by 3, I swap boys. After swapping, now what I will do,
21:43here boys, either multiply by 3 on both side or send the 3 to right-hand side, the division
21:55changes to multiplication, so it becomes 3 multiply with x, 3x is equal to 2y plus 4,
22:04then I send 4 on the other side, that is 3x minus 4 is equal to 2y, then what I will do,
22:21I will divide by 2 on both the side boys, left-hand side and right-hand side, rule,
22:29there is a rule in maths, if you divide left-hand side by a number, with same number you divide
22:34right-hand side, so 2 and 2 get cancelled, so y, I can replace y, f inverse of x is equal to 3x
22:44minus 4 by 2, I got the same thing. So, boys, now one more thing, the flow diagram,
23:03so flow diagram for f, first I will write for f, so or you can say g, whatever you take,
23:21suppose g at x whatever you have taken, g at x is 2x plus 4 by 3, so boys, so first I will write
23:38for the function for g, that is x, then multiply by 2, here we write multiply by 2, so what we are
23:52doing, we are multiplying x by 2, so it become 2x, in this we add 4 boys, add 4, so now it become
24:04boys 2x plus 4, first you are adding x, you see here, 2x plus 4 by 3, so first what I have taken,
24:26I have taken x, then in this I multiply by 2, so it become 2x, in this I add 4, I added 4,
24:49so it become 2x plus 4, 2x plus 4, now after addition, then what I am doing boys, I divide,
25:01I divide by 3, so it become 2x plus 4 by 3. Now boys, I will go the reverse way now,
25:14okay, what I will do, I will go in reverse order, same but see here, what I will do,
25:23I will take x from right hand side, take an x, so divide, okay, first in place of divide,
25:34I will change it to multiply, because the inverse of division is multiplication,
25:40so I will write multiply by 3, so it become 3x boys, after that there is add is there,
26:00so you see this function add, so I will subtract now, subtract 4, so it become boys 3x minus 4,
26:17okay, now last step, here multiply by 2, this is very important boys see, multiply by 2,
26:31so here I will write divide by 2 after subtraction 3x minus 4 by 2, so I got the inverse function 3x
26:42minus 4 by 2 at last, so I was writing x, so g at x was 2x plus 4 by 3 and g inverse at x,
27:03what we got by different ways, 3x minus 4 by 2, so here first step we have taken x,
27:19multiply by 2 first step, then add 4 and divide by 3, so remember divide by 3,
27:29so we have taken x on right hand side, divide by 3, we change into multiply by 3,
27:35so we got 3x, then add 4, we subtract 4, then multiply by 2, we divide by 2,
27:47so we got 3x minus 4 by 2, it is so simple boys, I hope you understand,
27:53thanks for watching the video, it is by RBA educational academy,
28:03please like, share and subscribe, so till the time we meet again, please share the video.