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Hello, and welcome to Bay College's online lectures.
For Math 105, Intermediate Algebra today, we're going to
conclude 8.2 in this video, part two.
Composite and inverse functions.
First thing we're going to look at is an example.
Here we have a function, and this is a function of our pay.
Let's say we make $10 per hour.
So if I work so many hours, I'll know how much pay I get.
So this is my pay as a function of hours. $10 per
hour is my rate.
So let's say I work 20 hours.
Let's look at this and say OK, what's my input?
20.
I work 20 hours this week at $10 an hour.
So if I evaluate p of 20, I get 10 times 20 is $200.
So this is our function, and we have our input and our
output, our x and our y.
But what if I didn't keep track of my hours, and I look
at my check, and before taxes it happens to be $200.
And I want to know well, how many hours did I work?
I would need to undo this math.
And in order to undo that, I say OK, well this
is the input I have.
$200 is the value of my check before taxes.
How many hours did I work?
Well, if I undo the math, instead of multiplying by 20
to get 200, or excuse me, multiplying 10 by 20 to get
200, I'm going to divide 200 by 10b.
A super cool operation.
Divide by 10, which wold give me 20.
So my input, my x value in this case was 200, and my
output was 20.
If you look at these two points, essentially what we
have is an inverse.
And we can apply this to functions.
If we look at our function machine here, our function f,
my input is x.
So I put x into the function, and my output is f of x.
Input of x, you can see here point, x, and my output, my y
value, f of x.
x and y, or f of x using that notation.
The inverse function, and I'll just point this out a couple
of times in this video.
This negative 1 as an exponent is not really an exponent.
It's just notation.
This is the inverse exponent.
So we use that to denote that this is an inverse function.
This doesn't mean that we're going to do reciprocal
operation on it.
So when dealing with an inverse function just like we
had here, we switched the x's and y's.
We switched the input for output, and the
output became our input.
So what am I putting into this function?
Well in this case, the output of my function is now the
input of my inverse function.
And my output becomes x.
As you can see, we have f of x and x, my input and my output.
They're switched from what we have here.
This is an inverse function.
Before we actually get into the algebra of inverse
functions, the first thing we want to do is we want to
identify functions and their inverses, and how we do that
is dealing with relations.
Here we have a function.
Well, maybe it's a function.
This relation, I should have wrote relation here is the
series of points negative 3, negative 27, negative 2,
negative 8, negative 1, negative 1, 0, 0, and 2, 8.
If I want to find the inverse of this relation, well,
essentially I have to replace my y values for x values, and
my x values for y values.
And I can do that simply by doing this.
Replacing x's for y's, and y's for x's.
This is the inverse relation to that.
Now we want to ask ourselves a question.
Is this a function?
Is it?
Well, if we look at this, one way to tell a function is to
say does the x value ever repeat?
If you recall, we did that when we used the
vertical line test.
There's only one x value as we move that vertical
line across our graph.
Well, we don't have a graph here, but another way of doing
that is does the x value repeat? negative 3, negative
2, negative 1, 0 and 2 are my x values.
None of them repeat, so this is in fact a function.
Well, how can I tell if its inverse is a function?
Well now the x's are y's, and the y's are x's.
Well, apply the same thing to the inverse.
Does the x value repeat?
No.
So the inverse is also a function.
So we call this a one to one function.
This is a one to one function, which means, its inverse is
also a function.
Now, a quicker way to determine if the inverse is a
function, we can do that before we
even write these points.
If the function doesn't have repeating x values, its
inverse cannot have the repeating y value.
So we see negative 27, negative 8,
negative 1, 0 and 8.
Well, here, negative 27, negative 8, negative 1, 0, 8.
We could have found that information right from this
relation here.
Just say, well, if the y values don't repeat, and the x
values don't repeat, we have a one to one function.
So let's look at this.
Again, I should have wrote relation.
It is relation.
And we're going to ask, is it a function?
Well, let's determine that.
We have negative 3, 4, negative 1, 2, and 3.
All unique values.
So the x value doesn't repeat.
So this is in fact a function.
Now, is its inverse going to be a function?
We can tell that before we even find what the inverse is
through the y values.
We have 9, negative 2, 1, 4, and 9.
The 9 repeats.
So it's not unique.
It occurs for two different values of x.
Negative 3 and positive 3.
That tells me if we look at its inverse, 9, negative 3,
negative 2, 4, 1, negative 1, 4, 2, and 9, 3.
If we look at this inverse of this function, or this
relation, we can see that 9 does repeat.
So this is not a function.
So this is not one to one because its
inverse is not a function.
We had that repeating x value.
So let's go over here, and one more time, I want to just look
at the notation that we're using when we get
into inverse functions.
We're familiar with this notation.
F of x equals something.
The inverse function we denote, f inverse of x.
That's just the inverse exponent, but that doesn't
mean that we're going to do that math.
It's just notation.
Keep that in mind.
So f inverse of x.
That's what this notation means.
This is the inverse function.
So we talked about the vertical line test, and here's
some good news.
We look at the vertical line test.
We go this way, and we can see for each value x, it only
crosses the graph once.
Which means for each x, there is a unique y.
Now, there is another test that we can check its inverse
using this graph without having to graph a new
function, the inverse function.
We can use the horizontal line test.
This is not a one to one function.
It is a function because it passes the vertical line test,
but its inverse will not be a function.
Because notice the y value repeats.
When I drew this horizontal line, it crossed the graph in
two areas, or more than one area.
Saying the inverse is not a function.
Now let's look at this. f of x equals x cubed.
If we graph that, maybe we're familiar with
that, maybe we're not.
Let's plot some points.
If I put in 0, I get 0.
So that's a point on my graph for f of x.
If I plot, let's put it in 1 here.
1 cubed is still 1.
If I put in negative 1, negative 1 cubed is still
negative 1.
If I put it 2, 2 cubed is 8.
That's some point way up here.
And if I put in negative 2, negative 2
cubed is negative 8.
That's some point way down here.
So we can kind of see a pattern here.
The cubic function looks like that.
Now, if I want to graph its inverse, I have this point
here, which was negative 2, 8.
A little sloppy there, I apologize.
And maybe I had this point up here, which was, I'll put it
over here, 2, positive 8.
Negative 2, negative 8.
Positive 2, positive 8.
So we have some points there.
Well, if I want to graph its inverse, well, to find an
inverse we just flip the points, right?
We put the 8 here and the 2 there, and we put the 8 here
and the negative 2 there.
Well, that's what we can do to the graphics inverse.
If I know some points, I just have to flip the x and y
values, and then go ahead and graph it.
So let's graph these points.
This point was 1, 1.
Well, that's not going to change if we flip it.
This point was negative 1, negative 1.
That's not going to change when we flip it.
0, 0.
Well, we still have 0, 0 when we flip it.
But these points, 2, 8.
Well, I want to have 8, which is some point
here, way out here.
And 2, 8, 2.
This is the point 8, 2.
And if I'm going to graph this, the inverse of negative
2, 8 would be negative 2, negative 8.
Its inverse is going to be negative 8, negative 2.
Some value down there.
Now if I graph this, bring it up a little bit, make it
symmetrical.
We can see, how are these different?
Well, what's different about them is they are essentially
mirror images of each other.
But it's through a special line, and that special line is
this right here.
1, 1, 2, 2, 3, 3, 4, 4.
It's a linear equation, and it's actually
our identity function.
y equals x.
It's a mirror image.
If we look, if this was a mirror, the distance from this
point to that line is equal to the distance from this point
to that line.
They're equal distances.
So if I can pretend this was a mirror, I could say well, if I
look in the mirror, I'm gonna see its reflection.
And same thing here.
I apologize for not being to scale, but this would be equal
distances from the line y equals x.
So we can find an inverse by graphing it as well.
All right.
What if we want to find functions' inverses
algebraically?
Well, this isn't too bad.
It's just a little bit of algebra.
And at this point in math, we should be
comfortable with algebra.
Now, if I have this function, f of x equals 3x plus 7, and I
want to evaluate it for f of 10, what exactly am I doing?
Well, the first thing I do is I multiply 3 times 10 if I'm
evaluating the function to get 30.
30 plus 7 is 37.
So my input was 10, and my output was 37.
So I want to find its inverse.
I'm going to do that algebraically.
Well, we talk about inverses.
Well, why is it called the inverse?
Because we're going to do the inverse operations.
How would I undo 37 to get back to 10?
Well, I would subtract 7.
And let me just write it here.
37 minus 7.
And then divide by 3.
So instead of adding 7, multiplying by 10, I'm going
to subtract 7 and divide by 3.
Excuse me, I said multiply in my head.
I meant multiply by 3.
Divide by 3.
So we did the inverse operations.
And that's why it's called the inverse function.
It undoes it.
And if we actually do this, 37 minus 7 is 30.
30 divided by 3 is 10.
In this case, the inverse operation, our input was 37,
and our output was 10.
Now, let's actually do that.
Let's find this.
How do we go about doing that?
Well, if we want to find the inverse of a point, we switch
x's for y's and y's for x's.
Well, let's just write this in this manner here.
This is my input, that's my output. x and f of x, x and y.
Now what I can do to find the inverse algebraically is just
switch my values, x and y.
Put x where we have a y, and put y where we have an x.
So this is going to be my inverse function, but it's not
in the right format.
In order to have an inverse function, it has to be the f
inverse of x. x has to be the input.
So how could I do that?
Well, if I were to do it here, we just undid the math.
So I'm going to subtract 7 from both sides.
So I get x minus 7 equals 3y.
And then I'm going to divide by 3.
So I get y by itself.
Well, y equals x minus 7 over 3.
Input this operation will give me my function.
Well, that is the function inverse.
Once we flipped them, it became the inverse.
So now I can use this notation.
y is equal to this value, f inverse of x is the new y.
And x minus 7 over 3 is the inverse function.
So algebraically, we just switch x's a y's
and solve for y.
That is the gist of finding the inverse function
algebraically.
Switch your x's and y, solve for y.
And here's our inverse function.
Now let's graph these, and see if we can see that reflection
through y equals x.
If I know I'm going to graph a function and its inverse, the
first thing I'm going to do is I'm going to write in the line
y equals x.
Because I know that every function and its inverse are a
reflection through our identity function, y equals x.
If x is 1, y is 1.
That's why it's called the identity.
One identifies the other.
All right, so let's graph f of x.
3x plus 7 is a linear equation.
When x, let's just do it by a few points.
When x is 0, y is 7.
So I'm going to write 0, 7 as one point.
And let's say x is, let's say negative 2.
We'll go 1, 2.
Negative 2.
3 times negative 2 is negative 6 plus 7 equals 1.
All right, so here's two points.
That's enough to graph a linear equation.
So I'm going to go ahead and do my best to draw a nice,
straight line.
A little easier when we have the actual grid.
But there's my line.
And now I'm going to graph its inverse.
Well, I could do the same thing.
Plot some here.
Let's do some points.
Let's say x is positive 7.
Some value here.
So 7 minus 7 is 0.
0 divided by 3 is 0.
Well, what is this point?
When I plug in 7 I got, oh, 0.
Notice 0, 7, 7, 0, they're inverse points.
x's and y's are switched.
Now I'm going to pick a value for x.
Let's say, well, let's do 1, because that's going to match
up this point here.
Or negative 2, it would be one.
So I'm going to plug in 1 to this function.
1 minus 7 is negative 6.
Negative 6 divided by 3 is negative 2.
so when I plugged in 1, I got negative 2.
And now if I attempt to draw a straight line with this, that
is the inverse function.
So this is my f of x, and this is my f inverse of x.
And if we look at these, we can see 0, 7, 7, 0.
They are reflections through this line,
equal distances apart.
So what's above the line here is reflected below the
line y equals x.
Over here, we can see what's above the line over here is
reflected below the line over here.
So this is how we can actually visualize it.
y equals x is our tool that we can use when we have to graph
a function and its inverse.
So let's do one more example of that.
All right, we're given f of x equals x cubed plus 1, and
we're asked to graph its inverse.
Well first, let's graph f of x.
If I want to graph f of x, which is x cubed plus 1, we
can plot some points.
But I'm not going to do that.
I'm just going to graph it.
I know it's a cubic function.
Shift it up one, if you remember the application to a
previous chapter.
You know, this is like a k value.
It just shifts the graph up one.
And the cubic function looks like that.
So there's our cubic function.
So now I want to graph its inverse.
Well, I didn't plot any points, so, well, I did one.
I have 0, 1.
So I know 1, 0 is a point on my inverse.
I flip them.
Now, how can I graph the rest of it?
Well, let's draw in my y equals x.
It's straight as we need it.
And we can say all right, well now I can use this y equals x
as my reflection surface.
My mirror.
So if I look at it from this point, it starts to get close
to the line, and then it goes away.
So from this point I'm going to get close to the line, and
then move away.
So I drew its reflection.
And then here, what's above the line from our point is
getting closer to it.
So I'm just going to draw it so that it gets closer to it.
And notice at the line, they actually share a point.
Because here at its surface, it's no distance between them.
And what's below here is moving very slowly.
I can draw that away.
We can see what's here is a reflection to what's here
through y equals x.
So we can graph it.
If you can graph one of them, you can just reflect the
other, and you'll have its inverse.
What if we want to find this value, this equation?
We can do it algebraically.
So let's say we have y equals x cubed plus 1.
This is our original function.
I'm just replacing f of x with y.
Different notation.
Now to find the inverse, I'm going to replace
the y with an x.
And the x with a y.
And undo the math.
That's our whole goal.
So I'm going to get y by itself.
I'm going to subtract 1 from both sides.
And then to get rid of this third power cubed, what I'm
going to do is I'm going to take the cubed
root of both sides.
So the cubed root, that's minus one equals the cube root
of y cubed is just y.
This is my inverse function, so I'm
going to use the notation.
y is f inverse, and that is equal to the cubed
root of x minus 1.
So this is the algebraic expression of this line right
here, and that's its inverse.
All right, how can we check our work?
Other than visually doing it or algebraically doing it,
what if we're not graphing?
What if we're just asked to do it algebraically?
There is always a way to check our work.
So let's go over here.
Using a composite function to check for inverse functions.
In the previous video, we dealt
with composite functions.
What they are and how they work.
Well, when dealing with functions and their inverses,
if we take a composite function of an inverse and its
function, or the function and its inverse, they always equal
the same thing, x.
So this is one way to check it.
The input is our answer, no matter which
direction we do it.
Now before when we talked about composite functions,
order mattered.
Well, the only time it doesn't is when we're
dealing with inverses.
If the order doesn't matter, we're dealing with an inverse.
So the first thing we're going to do is we're going to use
one of these.
And it doesn't matter which one.
I'm going to choose this one right here.
F of f inverse.
So let's apply this composite function.
I'm going to take f inverse or f, 3 times x plus 7.
And I'm going to evaluate it for the whole function f
inverse of x.
So I'm going to take this whole thing and plug
it in for this x.
Now, if I evaluate this 3 times this quantity, minus 3's
can cancel out.
So I'm left with x minus 7 plus 7.
And x minus 7 plus 7, negative 7 plus 7 is just x.
So this does in fact equal x.
So I know that this function and this
function are inverses.
So I was correct to say f of x and f inverse of x.
So this has been section 8.2 of Intermediate Algebra.
Thank you for watching.