Investigation 2: Functions

So now you know that in a function, one input cannot be

assigned to more than one output.

But note that several x values can be assigned

to the same y value.

In this second diagram, what's important to see is that each

x is paired with exactly one y.

Next, some vocabulary and notation.

Suppose the function f assigned each domain value x

to the range value y equals 5x plus 1.

y equals 5x plus 1 is called the function rule or equation.

It defines the relationship between x and y.

In any function, the value of y depends on the value of x.

So we say y is a function of x, and we write

y equals f of x.

This is function notation.

So, in our example, y equals 5x, and f of x equals 5x are

equivalent.

Let's start by graphing four simple functions, and from

their graphical representation, intuitively

deduce the domain and range of each function.

Press the Home key to open a new document.

Decide if you wish to save the open one, and then press 2 for

a graphs and geometry page.

Under Page Layout, choose Select Layout and pick layout

2, the vertical split screen.

Press Control-Tab to move to the right.

Next, press Menu, and again, select 2.

We'll graph two functions on each of Cartesian planes.

The cursor is blinking by f1, so key in the letters I-N-T,

then the left paren, x, and finally the right paren.

For f2 type in negative x.

To zoom in, use the nav pad to move the pointer to the

first tic mark on the x-axis to the left of the origin.

When it becomes an open hand, press Control and click to

grab this point.

Use the left arrow to drag it all the way to the left edge

of the monitor.

Press Escape to exit grab and drag mode.

You can now see unit one on the x-axis.

The first function is called the integer part function.

Its graph is like a stair case, and often

called a step function.

It assigns the integer part of x to any real number.

The second is a linear function with

slope negative 1.

Negative slopes yield descending graphs,

from left to right.

Linear functions have a straight line graph, as you

may have guessed.

A vertical line test, from left to right, yields exactly

one intersection point on each graph for every value of x.

That confirms that we have two functions, and it also tells

us that in both cases, the domain and range

are all real numbers.

Indeed, int of x is defined for every x, since all real

numbers have an integer part, and the only one.

In particular, numbers between 0 and 1 have integer part 0.

And negative x is also defined for every real number, since

all numbers have an opposite.

For example, 1/2 has negative 1/2, and negative 5 has

negative negative 5, or positive 5.

For these two functions then, the domain, d, and range, r,

both equal the set of all real numbers.

Notice a thicker letter was a real number to distinguish it

from all other uses of the letter r, such as r for range.

Let's go back to the inspire and graph two more functions.

Press Control-Tab to move to the left.

Type in x squared for f, please.

For our fourth function, enter 1 divided by x for the

reciprocal function, 1 over x.

To zoom in this time, press Menu, and under Window, select

to zoom in.

A center box appears.

Press click twice, then Escape.

x square is a quadratic function, and the shape of all

quadratic function graphs is called a parabola.

This parabola lies in quadrants one and two, where

all y values are positive.

The two north-bound branches continue upwards to infinity.

A vertical line test yields exactly one intersection point

for every value of x.

1over x is called a rational function, and its graph is

called a rectangular hyperbola.

This hyperbola lies in quadrant one, where y is

positive, and quadrant three, where y is negative.

It has four infinite branches and nowhere does the graph

ever touch the x- or y-axis.

So, x and y are never 0.

A vertical line test yields exactly one intersection point

for every value of x, except 0.

Indeed, x squared is defined for all x, since all real

numbers have a square.

But the squares themselves, namely the y values, are all

positive, except 0 squared, which is 0.

So the domain is R, and the range is the set of all real

numbers that are greater than or equal to 0.

1 over x is defined for all real values of x,

except when x is 0.

A 0 in the denominator makes the fraction undefined.

So the domain of this function is all real numbers, except 0.

The same is true for the y value, as we saw that the

graph never approaches the x factors, which means y never

takes on the value of 0.

In separate episodes of this series, we look in more depth

at the different types of functions, including linear,

quadrtic, rational, exponential, and so on.

Another important concept is the notion of increasing and

decreasing functions.

Let's take a look.

f2 of x equals negative x is a decreasing function.

We can see that the graph descends from left to right.

This means that as x increases, y decreases.

f3 of x, on the other hand, is decreasing for negative values

of x, and increasing for positive values of x.

Finally, f4 of x is decreasing on two separate intervals, for

both negative and positive values of x.

I'll let you revisit f1 of x on your own.

I suggest you try graphing it by hand, on good paper, and

pay close attention to the endpoints of

each horizontal set.

We'll end this lesson with an exciting inspire exploration

that visually connects the circumference of a circle to a

function graph.

Press Control-N for a new document.

We're going to construct a circle.

Press Menu, and under shape, select circle.

Move the point toward the left of the monitor, and click to

place the circle's center.

Use the nav pad to create a circle, then click again.

Now, we measure the radius.

Press Menu, and under measurements, pick Length.

Move the cursor to the center, click, then drag it with the

down arrow until you reach a point on the circumference.

Click twice, and the radius length appears.

Next we measure the perimeter, or circumference.

With the nav pad, move to a point at the top of the circle

and click twice.

Now, the perimeter appears.

To name this variable, press the Var key and

select store Var.

Type the letter P for perimeter, then press Enter.

Do the same for the radius.

Hover over it, and likewise, press Var.

Choose 4 Var again, and this time type r for radius, and

then press Enter.

You've now labelled both variables.

We're ready to split the screen.

Under Page Layout, select Custom Split.

With the nav pad, drag the partition to the left, making partition to

the left screen about one-third the screen's width.

Then press Enter.

Press Control-Tab to move to the right. Menu to make a

selection. then 2 for a graph and geometry page.

You now have a plain geometry table on the left. and an

analytic geometry page on the right.

Since radii and perimeters, or circumferences in this case,

are positive, let's change the setting.

Press Menu. and under Window, select Window Settings.

Enter negative 2, 10, negative 4, then 50.

Click OK.

To place the point, press Menu, and under Points and

Lines, pick Point.

Click to plot a point.

Next, press Menu again, and under Actions, select

coordinates and equations.

Click twice to place the point's coordinates.

We now want to connect the x value with the radius and the

y value with the perimeter.

To do this, hover over the x-coordinate of the

point and press Var.

This time select Link To, and then the letter r.

Press Enter.

Do the same for the y-coordinate.

Press Var, Link To, the letter p, and then press Enter.

Finally, switch back over to the left screen by pressing

Control-Tab.

Move the pointer to a point on the circle.

When the word circle appears. click and

hold to grab the point.

And now for the amazing part.

Use the nav pad to drag the circle, making it very large

and then very small.

Notice that the point r-p travels along

an invisible graph.

What's the shape of this graph?

As the circle becomes very small. the point is closer to

the origin.

Why?

It looked like the point was tracing an ascending line.

starting at the origin.

A little algebra can help.

The perimeter of a circle is 2 pi r.

So the function rule is p equals 2 pi r.

That's approximately p equals 6.28r.

When r approaches 0. so does p.

When r grows very large. so does p.

r and p vary proportionally at a rate of 1 to 2 pi.

It's therefore. a linear function with slope 2 pi.

We've come to the end of our lesson.

I hope this dynamic approach to algebra has helped you

understand the meanings of relation, function, domain,

range, graph, and much more.

Oh wait, there's one more fun problem still to come.