It’s Professor Dave; let’s learn about functions.
The standard practice of teaching math in school is to teach a year of algebra first,
then a year of geometry, and then do another year of algebra, which is often called algebra two.
Since we’ve just done a lot of work with geometric figures, we are ready to come back
to algebra and expand our understanding of this subject.
The first thing we want to do is comprehend the idea of a function.
A function is something that relates two quantities, an input value, and an output value.
So we can think of it as a little box, and when we insert input values, it spits out
the output values, in a way that depends on what the function is.
Let’s say at a particular store, everything is thirty percent off today.
That means to find the new price of an item we would multiply the original cost by 0.7,
because that will give us seventy percent of the original price, which is the same thing
as thirty percent off, since one hundred minus thirty is seventy.
The function that we would therefore use to compute the sale price of any item would be
F of X equals 0.7 X. F is the function, and, the X in parentheses, which is read “of
X”, means that the function will operate on any X value that you plug in.
If we plug in one dollar for X, the function spits out seventy cents, so F of one equals 0.7.
F of two equals 1.4, F of three equals 2.1, and so forth.
We could make a table like this, and we could even graph the resulting relationship.
So everything we used to do with equations where Y depends on X, we can do the same thing
with functions.
The only difference is that with functions, for any X value we plug in, there must only
be one Y value.
For example, if a function tells us the sale price of an item, if we plug in the regular
price, we should get only one sale price.
For that reason, for a graph to be representative of a function, it must pass the vertical line test.
Since a function can only have one output, or Y value, for each unique input, or X value,
then if we move a vertical line across a function, it will only intersect that function at one
point, wherever the vertical line may be.
If the vertical line intersects at two or more spots, it is not a function.
That would be as though we plugged in the regular price of an item and got back two
different numbers for the sale price.
We can, however, have the same output for multiple inputs, which is why different X
values are permitted to produce the same value for the function, so there is no horizontal
line test necessary.
This line is a function.
This circle is not.
This curve is a function.
This curve is not.
We can also describe the domain and range of a function.
The domain is essentially all of the values that can be plugged into the function.
So if we have something simple like F of X equals two X plus one, the domain would be
all real numbers, or negative infinity to positive infinity.
All the function does to the input is double it and add one, so any number will work.
But there are examples where there are limitations to the domain.
Let’s say F of X equals one over X minus two.
We can’t divide something by zero, so X can’t be two.
If X was two, we would get zero on the bottom, and the function would be undefined, so two
is not part of the domain.
The range, on the other hand, is all the potential output values, or essentially the values that
the function can possess.
Again, something simple like F of X equals two X plus one will yield a range that includes
all real numbers, extending to infinity in each direction.
But if we have something like F of X equals the absolute value of X, then the range is
now limited to values greater than or equal to zero.
So while the domain is very frequently all real numbers, the range has a tendency to
vary quite a bit depending on the type of function we are looking at.
We will also sometimes want to identify the zeroes of a function.
These are the input values that make the function equal to zero, which we can identify graphically
as the X-intercepts.
Wherever the function crosses the X-axis, that value of X is producing a Y value of
zero, so that is a zero of the function.
Again, different types of functions will have different numbers of zeroes.
Before we look at different types of functions, we need to make sure we know how to evaluate
a function.
For the function three X squared minus five X plus two, what is F of two?
Well let’s rewrite this but with two in the parentheses instead of X.
That means we are evaluating F of two.
We then also put a two instead of X wherever there is an X.
Two squared is four, times three is twelve, five times two is ten, so twelve minus ten
plus two equals four.
F of two equals four.
Now that we know what functions are, let’s check comprehension.