Graphing and transforming functions

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Exponential functions and identities

Let’s discuss the exponential functions by starting with the basics, the exponential identities.

Exponentials are fun, and for starters, it is enough to know that

but don’t panic, there won’t be anything devilish here.

The first exponential identity tells us what happens if we multiply this by - let’s say - 62.

Well, let's check it out.

If we multiply these, then

the exponents will be added up.

This is the first identity.

EXPONENTIAL IDENTITIES

Let's see what happens if we divide these.

But there is a minor inconvenience.

Here comes.

Well, when the exponent of the denominator is greater, then the result is a fraction.

Here the exponent will be negative.

Next, let's see how to raise a power to a power.

Well, like this:

We have to multiply the exponents.

Then, we have this thing:

So, what could this be?

Let's see what happens if we apply our newest identity to this.

So, it is something whose square is 9.

There happens to be such a thing, we call it .

So, a fractional exponent means a radical.

Developing the previous two identities a bit further, we get a third one.

If we have something like this:

then we can try applying this formula.

Here come a few more formulae,

but now, finally, let’s see the functions.

The function of 2x looks like this: This one is 3x.

If the base number is between 2 and 3, then the function will be between 2x and 3x.

For example, one such number is

2.71828182845904523536028747135266249775724709369995…

This number has a powerful significance in mathematics, and to simplify things, it is called “e”.

Thus this function is ex.

All exponential functions with a base greater than 1 look something like this.

If the base is less than 1, well, that is an entirely different animal.


Logarithmic functions and identities

Here comes the logarithm

Now a new character enters the scene, the logarithm.

Well, this logarithm is a splendid thing, but requires a bit if an explanation.

All it is, really, that  tells us to which power should “a” be raised in order to get x.

Here is this one, for instance:

This means that we have to find to which power we should raise 2, so that we get 8.

Well, 23=8, so the answer is...

Or let’s see this one:

Well, let's see

Here comes a bit more difficult case:

The question is how 8 would turn into 2. Dividing by 4 is not a good answer here, because we are looking for something exponential.

The correct answer is:

Let’s figure out what this is:

The question is: 8 raised to which power would result in 16?

Well, there is something common in 8 and 16, namely 2, because 23=8 and 24=16.

Then, we get from 8 to 16 by first turning 8 into 2,

and then from 2 we make 16.

After all these, this should not be a problem:

Neither this one:

And now let's see the logarithmic identities.

LOGARITHMIC IDENTITIES

One of the biggest benefits of the logarithm is that we can solve equations like this:

We take the logarithm of both sides.

And Voila!

In general, if we have something like this: 

then we can get x by doing this:

We should remember the reverse of it, too. If

then we can get x by doing this:

Solving exponential equations

Solving logarithmic equations

Let’s solve these, for instance:

And now let's see the functions.

Well, the logarithm is only defined for positive x values.

If the base is greater than 1, then the function is increasing.

If the base is less than 1, then it is decreasing.


Trigonometric functions and identities

Let’s talk a bit about trigonometric functions.

Well, here is a circle with a unit radius.

In this circle, the first coordinate of the unit vector with a directed angle  is ,

and the second coordinate is .

 and  are periodical functions.

This means they repeat themselves at certain intervals.

This interval is called a period, and in their case, the period is .

If we have an equation like this:

well, this, due to the periodicity, should have infinitely many solutions.

Furthermore, there is a blue solution,

and there is a green one.

This is produced by the calculator,

and this is the period.

Well, the calculator won’t spit this one out, so we have to use a little trick.

The sine function works so that there is always a blue solution that the calculator supplies,

and there is a green that we get

so that their sum is always .

It pays to remember this.

Let's see what it looks like for cosine.

Here, we will have a blue and a green solution,

infinitely many of each, to boot.

The situation is a bit simpler than in the case of the sine, as the blue and the green solutions are always the negative of each other.

The blue is produced by the calculator,

and if we insert a negative sign in front of it,

well, then we have the green one.

So the cosine is a lot better than the sine.

And now come some other species.

Let’s see how these look.

Well, not very pretty.

We may be able use them as wallpaper patterns, but that’s about it.

After such visual delights, here comes a deluge of trigonometric formulae.

We will only look at the first one million most important formulae.

THE MOST IMPORTANT TRIGONOMETRIC RELATIONSHIPS

Here, inside a unit circle, there is a right triangle,

for which we apply the Pythagorean formula.

Well, this may be the single most important trigonometric relationship.

It has two mutant versions.

Now, we will see more tricks in the unit circle.

And here come a few more.


The inverse function

Each function is a  assignment, the inverse of which, if it exists at all, is the  reverse assignment.

Calculating the inverse goes like this:

Let

First, write the function in the form of y=thingy:

Here, we assign y to x.

The inverse is the reverse assignment, where we assign x to y. So, the purpose is always to rearrange y=thingy to x=something.

Finally, we swap x and y (some people don’t do this), and then we get the inverse.

The inverse is denoted by:

But, there is a little trouble. Not all functions have an inverse, as not all assignments can be reversed.

For example, in the case of , we have  and , and thus, we cannot reverse this: .

The crux of the problem is that this function assigns the same number to two different numbers (2 and -2), and therefore, the assignment cannot be reversed.

But if we exclude the negative numbers,

then everything will be all right.

So, only those functions have an inverse where for two different x values

there are two different y-s assigned.

We say they have a one-to-one correspondence, shortly, they are injective.

A function  is injective, if  then .

All strictly monotonic functions are injective and thus, invertible.

And there is one more thing here.

Let the function be  and its domain .

Well, then the image is .

The inverse function is the reverse assignment, so in this case, these are swapped.

If  is invertible, then its Domain is identical to the Range of its inverse, and its Range is identical to the Domain of the inverse.

Let's see a few examples!

Let’s find the inverse of function , if

There is no inverse, because the function is not injective.

For example, it assigns the same number to 4 and -4, namely 0.

This case is entirely different, as x could only be positive here. There are no two positive numbers with the same square, so this function is injective.

Let’s see the inverse:

In this case, there is an inverse, too, because the function is injective.

Let’s see the inverse!

In this case the function has no inverse, because it is - again- not injective. For example, it assigns the same number to 4 and -4, namely 0.

Unfortunately there is no inverse in this case either, as the function is not injective.

Let’s see one more.

Here is this function, and we want to find its inverse.

 and

Finally let’s see this one, too:

Let’s talk a bit about the geometric meaning of the inverse.

Here is a function

and let’s see what happens to the graph of this function when we invert it.

Well, this.

Let’s reflect the graph of the function about the y=x line.

It is clearly visible on the drawing that the inverse of radical functions is never a full parabola, only a half.

And the reverse is true as well: a full parabola can never be inverted, only its half.

Here comes another splendid function:

Well, the inverse of this function is:

The inverse of the exponential functions are the logarithmic functions.

And this is mutual: the inverse of the logarithmic functions are the exponential functions.

Let’s see the inverse of this, for instance:

We can lure the x out of the exponent by taking the logarithm of both sides.

Here is another one, for example:

Functions  and  are also inverses of each other.

We should be careful with this function inversion thing, as it could be harmful in large doses.

But maybe we can get away with one more...


The geometric meaning of the inverse

Each function is a  assignment, the inverse of which, if it exists at all, is the  reverse assignment.

Calculating the inverse goes like this:

Let

First, write the function in the form of y=thingy:

Here, we assign y to x.

The inverse is the reverse assignment, where we assign x to y. So, the purpose is always to rearrange y=thingy to x=something.

Finally, we swap x and y (some people don’t do this), and then we get the inverse.

The inverse is denoted by:

But, there is a little trouble. Not all functions have an inverse, as not all assignments can be reversed.

For example, in the case of , we have  and , and thus, we cannot reverse this: .

The crux of the problem is that this function assigns the same number to two different numbers (2 and -2), and therefore, the assignment cannot be reversed.

But if we exclude the negative numbers,

then everything will be all right.

So, only those functions have an inverse where for two different x values

there are two different y-s assigned.

We say they have a one-to-one correspondence, shortly, they are injective.

A function  is injective, if  then .

All strictly monotonic functions are injective and thus, invertible.

And there is one more thing here.

Let the function be  and its domain .

Well, then the image is .

The inverse function is the reverse assignment, so in this case, these are swapped.

If  is invertible, then its Domain is identical to the Range of its inverse, and its Range is identical to the Domain of the inverse.

Let's see a few examples!

Let’s find the inverse of function , if

There is no inverse, because the function is not injective.

For example, it assigns the same number to 4 and -4, namely 0.

This case is entirely different, as x could only be positive here. There are no two positive numbers with the same square, so this function is injective.

Let’s see the inverse:

In this case, there is an inverse, too, because the function is injective.

Let’s see the inverse!

In this case the function has no inverse, because it is - again- not injective. For example, it assigns the same number to 4 and -4, namely 0.

Unfortunately there is no inverse in this case either, as the function is not injective.

Let’s see one more.

Here is this function, and we want to find its inverse.

 and

Finally let’s see this one, too:

Let’s talk a bit about the geometric meaning of the inverse.

Here is a function

and let’s see what happens to the graph of this function when we invert it.

Well, this.

Let’s reflect the graph of the function about the y=x line.

It is clearly visible on the drawing that the inverse of radical functions is never a full parabola, only a half.

And the reverse is true as well: a full parabola can never be inverted, only its half.

Here comes another splendid function:

Well, the inverse of this function is:

The inverse of the exponential functions are the logarithmic functions.

And this is mutual: the inverse of the logarithmic functions are the exponential functions.

Let’s see the inverse of this, for instance:

We can lure the x out of the exponent by taking the logarithm of both sides.

Here is another one, for example:

Functions  and  are also inverses of each other.

We should be careful with this function inversion thing, as it could be harmful in large doses.

But maybe we can get away with one more...


Graphing and transforming functions

Let’s start with a very simple thing.

Let's see how functions work.

Well, here is the x axis, full of numbers.

x axis

The function takes certain numbers from these, and assigns another number to them.

For example, it assigns their square.

This function is written like this:

Most of the time we will use this third type of notation.

Those lucky x values for which the function assigns something, are called the Domain, denoted by .

For x2, this is the entire x axis.

But here comes, for instance,  ,

which is not defined for negative x values.

Therefore, the domain:

The part of the y axis that is assigned to the x values is called the Image, or the Range.

The Range is denoted by

And now, let’s get back to function x2.

The graph of function x2 is a parabola. The vertex of this parabola is in the origin.

But if we replace x with

then, well, the vertex moves.

The vertex of the parabola is always at the point where this is zero.

Right now it is at .

And then we have this, for instance.

If we add three outside the squaring function,

the graph will move up by 3 along the y axis.

This is called internal transformation,

and this one is external.

If there is such a thing as:

then, due to the internal transformation, it is shifted along the x axis,

and due to the external transformation, it is also shifted along the y axis.

Let’s see what happens if we write 2x here.

Well, then we see a bit of stretching along the y axis,

but this is not too exciting.

What is much more interesting is that the shifting has changed as well.

And now let’s see what this would look like.

We already know .

We have to move this along the x axis, let’s see...

by 3.

And by 2 along the y axis.

If there is such a thing as:

then, well, due to the 3x it will stretch somewhat,

and then the usual stuff.

If we write a negative sign in front of , then we will reflect the graph about the x axis.

If we insert the negative sign inside the radical, then we will reflect the graph about the y axis.

And if we feel like it, we can reflect the function about both axes.

It gets really exciting if we combine it with the shifting business.

Let's see, for instance, what this function would look like.

There will be some shifting along the x axis,

then along the y axis as well,

and finally, due to the negative sign, a reflection.

But if the negative sign is on the outside, then it is an entirely different case.

This is splendid; let's see what else could possibly come.

The art of “completing the square”.


The art of “completing the square”

We will try graphing this function based on the transformations shown in the previous slides.

In order to decide which transformations was this function a victim of, first we will have to do a very funny thing with it.

We call this procedure “completing the square”. We will need it many times in the future, so let’s get it over with.

It boils down to these two identities:

This time we are going to use them in this direction.

We will look at the function over and over, until we recognize one of the identities.

Let’s see, what could be the value of b?

Well, this:

And now we can graph this, if we remember the previous slides.

Let’s see this one, too:

Now it is time to discuss some new functions.

In the next slideshow there come the exponential functions.