1
00:00:00,000 --> 00:00:00,830


2
00:00:00,830 --> 00:00:05,770
So we have f of x is equal to negative x plus 4, and f of x

3
00:00:05,769 --> 00:00:08,459
is graphed right here on our coordinate plane.

4
00:00:08,460 --> 00:00:11,690
Let's try to figure out what the inverse of f is.

5
00:00:11,689 --> 00:00:14,769
And to figure out the inverse, what I like to do is I set y, I

6
00:00:14,769 --> 00:00:18,589
set the variable y, equal to f of x, or we could write that y

7
00:00:18,589 --> 00:00:22,129
is equal to negative x plus 4.

8
00:00:22,129 --> 00:00:25,039
Right now, we've solved for y in terms of x.

9
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To solve for the inverse, we do the opposite.

10
00:00:26,750 --> 00:00:29,600
We solve for x in terms of y.

11
00:00:29,600 --> 00:00:31,570
So let's subtract 4 from both sides.

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You get y minus 4 is equal to negative x.

13
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And then to solve for x, we can multiply both sides of this

14
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equation times negative 1.

15
00:00:41,960 --> 00:00:47,530
And so you get negative y plus 4 is equal to x.

16
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Or just because we're always used to writing the dependent

17
00:00:50,390 --> 00:00:52,770
variable on the left-hand side, we could rewrite this as x is

18
00:00:52,770 --> 00:00:55,620
equal to negative y plus 4.

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00:00:55,619 --> 00:00:58,299
Or another way to write it is we could say that f

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00:00:58,299 --> 00:01:06,840
inverse of y is equal to negative y plus 4.

21
00:01:06,840 --> 00:01:09,469
So this is the inverse function right here, and we've written

22
00:01:09,469 --> 00:01:13,359
it as a function of y, but we can just rename the y as x

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00:01:13,359 --> 00:01:14,950
so it's a function of x.

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00:01:14,950 --> 00:01:16,079
So let's do that.

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00:01:16,079 --> 00:01:23,120
So if we just rename this y as x, we get f inverse of x is

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00:01:23,120 --> 00:01:25,780
equal to the negative x plus 4.

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These two functions are identical.

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Here, we just used y as the independent variable, or

29
00:01:30,730 --> 00:01:31,760
as the input variable.

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00:01:31,760 --> 00:01:34,660
Here we just use x, but they are identical functions.

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Now, just out of interest, let's graph the inverse

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function and see how it might relate to this

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00:01:40,219 --> 00:01:42,189
one right over here.

34
00:01:42,189 --> 00:01:44,250
So if you look at it, it actually looks

35
00:01:44,250 --> 00:01:45,510
fairly identical.

36
00:01:45,510 --> 00:01:47,010
It's a negative x plus 4.

37
00:01:47,010 --> 00:01:48,395
It's the exact same function.

38
00:01:48,394 --> 00:01:51,629
So let's see, if we have-- the y-intercept is 4, it's going

39
00:01:51,629 --> 00:01:52,949
to be the exact same thing.

40
00:01:52,950 --> 00:01:56,810
The function is its own inverse.

41
00:01:56,810 --> 00:01:58,990
So if we were to graph it, we would put it right

42
00:01:58,989 --> 00:01:59,929
on top of this.

43
00:01:59,930 --> 00:02:02,560


44
00:02:02,560 --> 00:02:04,730
And so, there's a couple of ways to think about it.

45
00:02:04,730 --> 00:02:07,530
In the first inverse function video, I talked about how a

46
00:02:07,530 --> 00:02:10,390
function and their inverse-- they are the reflection

47
00:02:10,389 --> 00:02:12,029
over the line y equals x.

48
00:02:12,030 --> 00:02:14,219
So where's the line y equals x here?

49
00:02:14,219 --> 00:02:16,444
Well, line y equals x looks like this.

50
00:02:16,444 --> 00:02:20,539


51
00:02:20,539 --> 00:02:25,709
And negative x plus 4 is actually perpendicular to y is

52
00:02:25,710 --> 00:02:27,780
equal to x, so when you reflect it, you're just kind of

53
00:02:27,780 --> 00:02:29,830
flipping it over, but it's going to be the same line.

54
00:02:29,830 --> 00:02:32,670
It is its own reflection.

55
00:02:32,669 --> 00:02:34,469
Now, let's make sure that that actually makes sense.

56
00:02:34,469 --> 00:02:38,699
When we're dealing with the standard function right

57
00:02:38,699 --> 00:02:43,479
there, if you input a 2, it gets mapped to a 2.

58
00:02:43,479 --> 00:02:48,750
If you input a 4, it gets mapped to 0.

59
00:02:48,750 --> 00:02:50,370
What happens if you go the other way?

60
00:02:50,370 --> 00:02:54,460
If you input a 2, well, 2 gets mapped to 2 either

61
00:02:54,460 --> 00:02:55,870
way, so that makes sense.

62
00:02:55,870 --> 00:02:59,180
For the regular function, 4 gets mapped to 0.

63
00:02:59,180 --> 00:03:02,319
For the inverse function, 0 gets mapped to 4.

64
00:03:02,319 --> 00:03:03,709
So it actually makes complete sense.

65
00:03:03,710 --> 00:03:04,610
Let's think about it another way.

66
00:03:04,610 --> 00:03:07,770
For the regular function-- let me write it explicitly down.

67
00:03:07,770 --> 00:03:09,390
This might be obvious to you, but just in case it's

68
00:03:09,389 --> 00:03:11,949
not, it might be helpful.

69
00:03:11,949 --> 00:03:14,439
Let's pick f of 5.

70
00:03:14,439 --> 00:03:18,020
f of 5 is equal to negative 1.

71
00:03:18,020 --> 00:03:23,900
Or we could say, the function f maps us from 5 to negative 1.

72
00:03:23,900 --> 00:03:27,230
Now, what does f inverse do?

73
00:03:27,229 --> 00:03:31,189
What's f inverse of negative 1?

74
00:03:31,189 --> 00:03:33,324
f inverse of negative 1 is 5.

75
00:03:33,324 --> 00:03:36,199


76
00:03:36,199 --> 00:03:41,000
Or we could say that f maps us from negative 1 to 5.

77
00:03:41,000 --> 00:03:44,139
So once again, if you think about kind of the sets, they're

78
00:03:44,139 --> 00:03:46,369
our domains and our ranges.

79
00:03:46,370 --> 00:03:49,069
So let's say that this is the domain of f, this

80
00:03:49,069 --> 00:03:50,759
is the range of f.

81
00:03:50,759 --> 00:03:59,039
f will take us from to negative 1.

82
00:03:59,039 --> 00:04:00,959
That's what the function f does.

83
00:04:00,960 --> 00:04:04,920
And we see that f inverse takes us back from negative 1 to 5.

84
00:04:04,919 --> 00:04:09,719
f inverse takes us back from negative 1 to 5, just

85
00:04:09,719 --> 00:04:12,319
like it's supposed to do.

86
00:04:12,319 --> 00:04:15,199
Let's do one more of these.

87
00:04:15,199 --> 00:04:19,069
So here I have g of x is equal to negative 2x minus 1.

88
00:04:19,069 --> 00:04:23,000
So just like the last problem, I like to set y equal to this.

89
00:04:23,000 --> 00:04:25,339
So we say y is equal to g of x, which is equal to

90
00:04:25,339 --> 00:04:27,739
negative 2x minus 1.

91
00:04:27,740 --> 00:04:29,949
Now we just solve for x.

92
00:04:29,949 --> 00:04:32,909
y plus 1 is equal to negative 2x.

93
00:04:32,910 --> 00:04:34,920
Just added 1 to both sides.

94
00:04:34,920 --> 00:04:39,050
Now we can divide both sides of this equation by negative 2,

95
00:04:39,050 --> 00:04:46,629
and so you get negative y over 2 minus 1/2 is equal to x, or

96
00:04:46,629 --> 00:04:52,420
we could write x is equal to negative y over 2 minus 1/2, or

97
00:04:52,420 --> 00:04:56,259
we could write f inverse as a function of y is equal to

98
00:04:56,259 --> 00:05:02,435
negative y over 2 minus 1/2, or we can just rename y as x.

99
00:05:02,435 --> 00:05:08,269
And we could say that f inverse of-- oh, let me careful here.

100
00:05:08,269 --> 00:05:09,250
That shouldn't be an f.

101
00:05:09,250 --> 00:05:11,300
The original function was g , so let me be clear.

102
00:05:11,300 --> 00:05:21,850
That is g inverse of y is equal to negative y over 2 minus 1/2

103
00:05:21,850 --> 00:05:24,340
because we started with a g of x, not an f of x.

104
00:05:24,339 --> 00:05:26,119
Make sure we get our notation right.

105
00:05:26,120 --> 00:05:31,009
Or we could just rename the y and say g inverse of x is equal

106
00:05:31,009 --> 00:05:34,319
to negative x over 2 minus 1/2.

107
00:05:34,319 --> 00:05:35,139
Now, let's graph it.

108
00:05:35,139 --> 00:05:37,969
Its y-intercept is negative 1/2.

109
00:05:37,970 --> 00:05:39,970
It's right over there.

110
00:05:39,970 --> 00:05:43,460
And it has a slope of negative 1/2.

111
00:05:43,459 --> 00:05:48,939


112
00:05:48,939 --> 00:05:52,759
Let's see, if we start at negative 1/2, if we move over

113
00:05:52,759 --> 00:05:56,500
to 1 in the positive direction, it will go down half.

114
00:05:56,500 --> 00:05:59,769
If we move over 1 again, it will go down half again.

115
00:05:59,769 --> 00:06:01,649
If we move back-- so it'll go like that.

116
00:06:01,649 --> 00:06:05,439
So the line, I'll try my best to draw it, will

117
00:06:05,439 --> 00:06:07,829
look something like that.

118
00:06:07,829 --> 00:06:10,579
It'll just keep going, so it'll look something like that, and

119
00:06:10,579 --> 00:06:13,169
it'll keep going in both directions.

120
00:06:13,170 --> 00:06:15,400
And now let's see if this really is a reflection over y

121
00:06:15,399 --> 00:06:21,909
equals x. y equals x looks like that, and you can see

122
00:06:21,910 --> 00:06:22,750
they are a reflection.

123
00:06:22,750 --> 00:06:25,439
If you reflect this guy, if you reflect this blue line, it

124
00:06:25,439 --> 00:06:27,219
becomes this orange line.

125
00:06:27,220 --> 00:06:30,885
But the general idea, you literally just-- a function

126
00:06:30,884 --> 00:06:34,459
is originally expressed, is solved for y in terms of x.

127
00:06:34,459 --> 00:06:35,529
You just do some algebra.

128
00:06:35,529 --> 00:06:38,750
Solve for x in terms of y, and that's essentially your inverse

129
00:06:38,750 --> 00:06:41,120
function as a function of y, but then you can rename

130
00:06:41,120 --> 00:06:43,699
it as a function of x.