It’s Professor Dave; let’s work with functions.
We just learned what functions are, and how to evaluate them.
But things can get a little trickier than this.
What if we want to plug in an algebraic term rather than a number?
For example, let’s say we have F of X equals two X plus one.
What is F of X plus three?
This works the same way as plugging in a number, we just put X plus three everywhere that we see X.
Two times quantity X plus three is two X plus six, plus the one, and we get two X plus seven.
We may also want to do algebra with two or more different functions, so we need to know
how to add, subtract, multiply, and divide functions.
Let’s say F of X is two X plus one, and G of X is X minus five.
We could do F plus G of X.
That’s the same as F of X plus G of X, which is two X plus one plus X minus five.
Combining like terms, we get three X minus four.
We could do F minus G of X, which is F of X minus G of X, or two X plus one minus the
quantity X minus five.
Don’t forget to distribute this minus sign, which inverts the signs of these terms, giving
us two X plus one minus X plus five, which becomes X plus six.
If we did G minus F, we would get a totally different answer.
We can also do F times G of X, which is F of X times G of X.
That would give us the product of two binomials, which we could FOIL.
We’ve already covered this method in detail, so hopefully we can do this quite easily to
get two X squared minus ten X plus X minus five, which we can then simplify by combining
the X terms.
And lastly we can do F over G of X, which is F of X over G of X.
In this case, we just put two X plus one over X minus five and that’s all we can do.
We should note that the domains of both F of X and G of X are all real numbers, and
that domain does not change for any of the above manipulations, except for the division
we just did, because with X minus five in the denominator, X can no longer be equal
to five.
We can also evaluate composite functions.
This is when we do something like F of G of X.
We will either denote this with a little open circle, or a little more logically, we can
put G of X where X goes in F of X.
This is different than F times G, because we are plugging G into the function F.
We are using the output of G as the input of F. In other words, we are taking F of X
minus five.
We plug in X minus five for X, multiply through by two and add one to get two X minus nine.
We can also find G of F of X, and it will be totally different.
We put two X plus one in for X in the G of X expression, and all we need to do is then
subtract five to get two X minus four.
But it doesn’t end here.
We could do F of F of X and G of G of X if we wanted.
This is not the same thing as squaring the function, which we can also do.
F of F of X would mean plugging in two X plus one where X goes, and that means multiplying
through by two and then adding one to get four X plus three.
We could do F of F of F of X, or G of F of G of X, or any sequence you can imagine, using
two or even more functions.
Now that we know how to work with functions, let’s check comprehension.