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This is going to be the third video about inverse functions, and in this video
I want to talk about
drawing the graphs of inverse functions.
So
let's start with this graph
of a simple linear function.
Here I've got the function y=2x-2,
and I've drawn the graph.
Now, to find the inverse of this function
we can do something really pretty simple.
Let's take a look at these two points where I've got the vertical intercept and the
horizontal intercept.
At the vertical intercept,
I can write the ordered pair
(0, -2).
And at the horizontal intercept I can write the ordered pair
(1, 0).
And, remember, in an ordered pair, the first number
is
the independent variable value, or the x-value, and the second number is the y-value,
the value for the dependent variable.
Now let's think about it.
If I'm going to do the inverse function, the graph of the inverse function,
what it means is I'm going to be swapping my x- and my y-values.
So
if I've got a point (0, -2) for my original function
that means in my inverse I'm going to have the point
(-2, 0),
(-2, 0),
which would be right here. I do that in red.
(-2, 0).
(-2, 0).
and if I've got this point on the horizontal
axis, (1, 0),
then on the vertical axis I'm going to have the inverse of that point,
(0, 1).
So (0, 1) will be right here.
And now, since this is a linear function,
which is a straight line,
I can just take a ruler
and connect those two points
and
I'll get the graph
of the inverse of that function.
So the black line is the original function
and the red line is the inverse.
And that's all there is to a simple linear function.
Now let's take a look at something a little more complicated.
Here I've got a function which is a jagged line, and I want to draw
the inverse
graph for this function.
So let's see.
I'm given these points on the graphs I've got a point
(-3, -2).
That means my inverse
is going to be,
is going to have the point
(-2, -3).
So let's see where that is. -2 is over here, and -3 is down here,
So I'm going to have
this point here
at (-2, -3).
All I did was take an ordered pair
and reverse the numbers.
I found the inverse for that ordered pair.
Then I've got this ordered pair (-2, 0), which means I want
(0, -2)
That's right here on the vertical axis
(0, -2).
(0, -2).
I can connect those two line.
And now, going on,
I've got this point here in the original function (0, 1). So I want
(1, 0).
That's going to be right here
on the horizontal axis.
So I'll connect this line further.
And then I go up to (2, 3), which means I want (3, 2).
So here's 3,
and 2,
so I'll go
up like this.
So the black line, once again, is the original function
and
the red line
is the inverse of that function. Now in general, this is a pretty sloppy graph, but if
you draw a line
where you have
y = x,
which is just a diagonal line,
what you're going to find
is that
an inverse function
is basically what you would get if you flipped your original function
along that dotted line.
If I were to fold this piece of paper in half
along that dotted line,
that black regional graph
should touch
the red
inverse line,
assuming I drew it nicely, which... this isn't really a great graph.
Now let's look at one more function.
You remember in the previous video we talked about
the horizontal line test?
And we had this parabola, y = x-squared,
and we saw that that parabola failed the horizontal line test.
Let's take a look at what would happen if we tried to graph it anyway.
So since it's y = x-squared,
when x is 1,
well x-squared is 1, so I'm going to have a point
(1, 1).
And when x is -1,
x-squared would also be 1, so I'm gonna have
(-1, 1).
And when x is 2,
well, 2-squared is 4,
so I'm going to be up here,
(2, 4).
And when x is -2,
I'm also up at 4. I've got
(-2, 4).
So
let's see what happens if we want to
take the inverses for those points.
Oh, and I've also got the point (0, 0),
since the graph goes right through the origin. So I know I'm gonna have a point
here
on my new graph.
And
if I invert this (1, 1), I still get a (1, 1).
If I invert (2, 4), I get (4, 2).
So that would be
around here somewhere.
So I can draw
the inverse of this curve here. It would
look something like this.
But when I go to draw the other half the of the graph, I've got (-1, 1),
so the inverse of that would be (1, -1),
And I've got
(-2, 4)
so the inverse of that would be (4, -2).
So 4 is here,
and -2 is here,
(4, -2),
and when I connect those dots,
well what I've done,
if I draw this diagonal line,
what I've done is like the equivalent of flipping this original parabola
across that diagonal line.
But now you can see that that fails
the vertical line test.
So the inverse is failing the vertical line test
because the original failed
the horizontal line test.
Take a look and think about the relationship between
these two graphs
and the idea of a horizontal line test for the black graph
and a vertical line test
for the red graph, and you can see why if you fail
the horizontal line test for your original graph
your inverse graph
will not be a function because will fail the vertical line test.
But, basically, assuming you've got a graph which is a function which is one-to-one,
which you can invert and get a function,
all you're going to do
is you're going to take points on the graph, take any point on the graph
that you know,
take the inverse of those points. In other words, take the ordered pair
and swap the numbers around,
and then plot those new ordered pairs,
connect your dots,
and you'll have a graph of the inverse.
So that's it for now.
Take care. I'll see you next time.