In class today we talked about all things related to
real-valued functions.
1. Introduction
Definition 1.1. A real-valued function f : D→ R is a rule which
as signs to each input in the set D an output
in the set of real numbers R.
There are a wealth of basic real-valued functions that you have seen previously
in your life. These include:
• polynomials (like
or
)
• rational functions ( quotients of polynomials , such as
or
)
• algebraic functions (functions that can be written by adding, multiplying,
dividing, or extracting roots
of polynomials, such as
• trigonometric functions (these are sin (x), cos (x), tan (x), csc (x), sec (x), or
cot (x))
• exponentials functions (such as ex or 2x)
• logarithmic functions (such as ln(x) or
)
• the absolute value function (written |x|)
2. Combining functions
Although there are lots of ‘basic’ functions we could talk about, we can create
really interesting functions by
combining those functions found in our library above. There are a few ways to
combine functions. For instance,
we can take two old functions and define a new function by adding them together.
The sum of two functions f
and g is defined by the rule
(f + g) (x) = f (x) + g (x).
Similarly we can take the product of two functions by multiplying their outputs
(f g) (x) = f (x) g (x).
A far more subtle way to combine functions is function
composition, where we evaluate one function ‘at’
another function. The composition of f and g is written as f o g, and is defined
by the rule
Example. Let f (x) = x2 +1, g (x) = sin (x), and h (x) = ex. What are f +h, f h, and
f o h? What is h o f o g?
Solution. The function f+ h is defined by the rule (f+ h) (x) = f (x)+ h (x), and of
course f (x)+ h (x) = x2+1+ex.
Similarly f g is defined by the rule (f g) (x) = f (x) g (x), so that (f g) (x) = (x2 +
1)ex.
The function f o h is given as
Finally, the function h o f o g is
It is also critical in this class that you are able to
take a given expression and identify it as a composition of
functions. When doing this, it’s good to work from the outside in.
Example. Write 
as a composition of functions.
Solution. The answer is 
To see this, I start by asking ‘What is the last thing I do when evaluating this
function?’ In this case, the
last thing to do is evalue the sin of some number, and therefore sin(x) will be
the last function I compose with
(i.e., it goes on the far left). Then I forget about the sin in my expression
(so that it becomes 
and
ask ‘What is the last thing I do when evaluating this function?’ The answer is
now that I raise 2 to a paricular
power, so that the function 2x is the second-to-last function in my composition
(i.e., it goes to the right of
sin(x)). Just as before, I now forget about the 2 and look only at the remaining
function tan(x + 1), and I ask
‘What’s the last thing I do when evaluating this function?’ Here the answer is
that I evaluate the tangent of a
number, so that tan(x) belongs in my composition. This leaves me with only x+1,
which goes on the far right
of my expression.
Note: This is easier to see me do in person, so if you want to see this again
just ask!
3. Function inverses
Now that we know how to combine functions, it is useful to know how to
‘uncombine’ them. That is, if I’ve
put f and g together to form a new function, how can I recover either f or g
from the new function?
If I want to recover f from f +g, we simply subtract g from f +g, just like
you ’d expect. Similarly if we want
to recover f from f g, we just divide by g. But how do I recover f from f o g?
This is a far subtler question,
and leads us to the concept of the inverse of a function.
Definition 3.1. For a function f and a value a in the
range of f, f -11(a) is the set of solutions to f (x) = a.
In other words, f -1(a) is the collection of all those inputs whose output is a.
Example. Let f (x) = x2. What is f -1(4)?
Solution. We know that f -1(4) is the set of all solutions to the equation f (x) =
4. Since f (x) = x2, this means
we are looking for solutions to x2= 4. Of course there are just the values x =
±2, so that f -1(4) = {−2, 2}.
This example is an important one to keep in mind, because it shows us that the
inverse of a function need
not be a function, since for some a in the range of f it may be the case that
f -1(a) is a set with more than one
element.
Frequently we will be interested finding the inverse of a function for an
arbitrary input. In order to do this,
first write y = f (x). Now switch the place of y and x in the expression, and
solve this new expression for y.
This will give you the inverse of your function.
Example. Let f (x) = 2x + 1. What is f -1(x)?
Solution. We are given y = f (x) = 2x + 1. To solve for f -1(x) we are first
supposed to switch y and x in this
equation, so we have x = 2y + 1. Now we solve for y in this expression, and the
result gives us f -1(x). Using
basic algebra techniques provides
Example. Let f (x) = e2x. What is f -1(x)?
Solution. Again, since our equation is y = e2x we are meant to solve for y in
the expression x = e2y. To do
this we take natural logs of both sides. On the left this gives ln(x), and on
the right this gives 
. But
and so we are left with ln(x) = 2y. Hence we have
The nice property of the inverse of a function is that for a function f, if its
inverse is also a function then
(when f -1(x) isn’t a function, we can only say
Because
this is true, we can use inverses to ‘undo’ function composition: if we want to
recover g from f o g, we simply
compose with f -1:
Similarly to recover f we precompose with
Of all the inverses we talk about, the two that will be important to keep in
mind are (in order of importance)
that ln(x) and ex are inverse functions (and similarly
and ax are
inverses for any a > 0); and that sin(x)
and arcsin (x) (or cos(x) and arccos (x), or tan (x) and arctan (x), etc) are
inverse functions.
4. Graphs of functions
You’ll be expected to know the graphs of familiar functions, including (but not
limited to) the graphs of a
line, parabola , cubic, sine, cosine, tangent, exponential function, and
logarithmic function. Given the graph of
a function f(x) you will also be expected to be able to sketch the graph of f (x
+ 1), f(2x), 2f(x), etc.
Given a graph, you can de termine if it is the graph of a function using the
vertical line test as follows. If
you are able to find a vertical line which hits the given graph in more than one
spot, than the graph fails the
vertical line test and is not a function (why? because there is an input which
yields two outputs). If there is no
vertical line which hits the graph in more than one spot, than you have a bona
fide function.
