And the ideas that up here,

x is in the x spot.

Let's figure out a table of values and see what that might look like.

If x is zero,

this is x and this is f(x),

if x is zero, then we have two to the zero, which is one.

and here's the 0.01. X is one,

we have two to the one equals two,

x is two, we have two to the two which is four,

x is three, two cube = eight.

Shoots all way up there.

X is minus 1, 2 to the minus 1 is 1 over 2 to the 1 is 1/2, and so on.

So the graph actually end up looking like this sort of, and on, and on.

So there's f(x) = 2 to the x. That's the graph f(x) = 2 the X.

Turns out that's strictly increasing, as you could probably tell.

Okay, in blue, let's figure out the graph of g(x) = 3 to the minus x.

So let's make a own table of values.

here's x, here's g(x),

there's 0, so g(0) would be 3 to the 0 is 1.

So it's here.

1 will be 3 to the minus 1.

1/3 and we get the pattern,

it's going to go quite steeper,

and up like that and down like that.

So, g(x) is strictly decreasing.

Okay. Let's draw the graph of h(x) = x squared in yellow.

So let's see that should go through the point 1, 1 and 1 minus 1,

looks about like that.

Now h is neither strictly increasing nor strictly decreasing.

Sometimes is going up and sometimes is coming down.

Here's a statement which you should intuitively understand what this means,

but h is strictly increasing on the interval from zero to infinity.

In other words, if I restrict myself only the things in this interval here,

and I only use those in inputs,

h satisfies the definition of strictly increasing.

I plug in two points in that- it goes up,

and h is strictly

decreasing on the interval from minus infinity to zero and you convince your self.

So that's interesting.

OK Let's understand what this might have to do with say, real world examples.

So we've seen some examples of functions that I've just made up,

here are some examples that might make more sense in the real world,

a world we might have increasing functions and decreasing functions.

On the left, let's imagine plotting over here years since birth of a typical child,

and over here, lets plot height.

We won't really commit to scale of units,

will just get a sense of the general shape.

So then of course nothing to the left of the y axis matters,

because they aren't negative years since birth.

This will almost entirely be an increasing function.

Right, you'll start at year one,

we'll measure you about here and you'll go up,

you go up really violently and then just start to level off.

Right around here that's probably about 17,

you'll stay sort of stable for a long, long time,

maybe a little bit of growth,

and you might get down a little bit sadly if you stoop.

So that's a function which is increasing for a long bit,

then it's sort of flat and then goes down.

As always the mathematical notions

will be much more precise and what happens in real life,

wouldn't be a smooth curve it would depend on the measurements of the doctor's office.

Over here, suppose we have years since purchasing a car,

and over here we have value of a car.

Without even drawing it, you probably predict that's going to be a decreasing function.

Right, here's year one,

it's one of my cars, it's going to start about a thousand dollars.

Perhaps that's revaluing me,

and then it's going to decrease.

Soon as you drive it off the lot goes down a little bit and it's going to

plateau somewhere near the bottom.

OK, we're going to close with

just a simple visual test to tell

whether or not our function is increasing or decreasing.

Notice the red graph is a graph of the strictly increasing function.

The blue graph is a graph of a function which is

neither strictly increasing nor strictly decreasing.

Really simple way to do it is called the horizontal line test.

You may notice, if I draw any on a line here in green,

it hits the right graph exactly once,

no matter where I draw the line.

Which makes sense, right.

Because that's the value that's hit that one time.

It could never, ever return to that value,

because if it did- the graph would have had to bend back down.

So in other words there's the value I hit,

and I don't get to hit that value again because the graph is gone.

See you later, it's all on the train.

On the other hand you'll notice blue,

there are many horizontal lines which hit the graph twice.

This horizontal line here hit here.

Now blue goes down,

again we've left the line forever but it starts to come back up again and hits the graph.

And so that's really the horizontal line test.

The function is strictly increasing or strictly decreasing if

whenever you graph it every single horizontal line has to intersect exactly once.