Download numbers, but f g is not. (4) Show examples when f g = g

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1. FUNCTIONS
(f −g)(x) = f (x)−g(x), and (f g)(x) = f (x)g(x). We have to be just a
little bit more careful with f /g, since this function is not defined when
g(x) = 0, even if x is in the domain of both f and g. So the domain of
f /g is the intersection of the domain of f and the domain of g, with
the exception of the points x satisfying g(x) = 0. For each point of this
domain, (f /g)(x) = f (x)/g(x).
If the range of f is part of the domain of g, then we can compose f
and g by first applying f and then g. The function we obtain in this
way sends x to g(f (x)) and is called the composition of f and g. It is
denoted by g ◦ f . Note that in g ◦ f , first f , and then g is applied.
Example 1. Let R be the set of all real numbers. If f and g are
both functions from R to R and f (x) = x2 and g(x) = x + 1, then
(g ◦ f )(x) = g(f (x)) = x2 + 1,
while
(f ◦ g)(x) = f (g(x)) = (x + 1)2 = x2 + 2x + 1.
Note that f ◦ g and g ◦ f are, in general, different functions.
3.3. Exercises.
(1) Sketch the graph of f (x) = x2 , g(x) = (x − 3)2 , and h(x) =
(2x + 5)2 .
(2) Sketch the graph of f (x) = cos 2x, g(x) = sin(x − 2), and
h(x) = 3 tan x.
(3) Show examples for f and g when g ◦ f is defined for all real
numbers, but f ◦ g is not.
(4) Show examples when f ◦ g = g ◦ f .
4. Viewing the Graphs of Functions
The graph of a function f is the set {(x, f (x))|x ∈ D(f )}. It is
a good way of visually describing what a function does. Today, we
have plenty of advanced tools, such as computer software packages and
graphing calculators, to study the graph of functions. In this section,
we point out a few of the common mistakes in using these tools.
In order to facilitate the discussion, let us agree on some terminology. If the domain of f contains an interval I and for all real numbers
x and x in I, it is true that x < x implies f (x) < f (x ), then we say
that f is increasing on I. Visually, this means that the graph of f goes
roughly from the southwest to the northeast while x ∈ I. Similarly,
if, for all real numbers x and x in I, it is true that x < x implies
f (x) > f (x ), then we say that f is decreasing on I. In terms of the
4. VIEWING THE GRAPHS OF FUNCTIONS
7
graph of f , this means that the graph goes roughly from the northwest
to the southeast.
If we simply ask a computer or graphing calculator to plot the
graph of a function without specifying the interval [x1 , x2 ] in which
the value of x can range, we may get an error message, or the computer may simply substitute default values for x1 and x2 . For example,
the software package Maple 13 uses the default values x1 = −10 and
x2 = 10. The interval [x1 , x2 ] is often called the viewing window.
We have to be careful, however, since not all viewing windows are
appropriate for all functions, and choosing an inappropriate viewing
window may cause misleading results.
For functions like f (x) = x, g(x) = |x|, or h(x) = x2 + 3, the viewing window [−10, 10] is appropriate as the behavior of these functions
outside that window is similar to their behavior inside the window.
Now let f (x) = (x + 10)2 . In this case, using the viewing window
[−10, 10], we get the graph of an increasing function. That is misleading
since f is decreasing on the interval (−∞, −10]. So, in this case, a
viewing window that starts at a point x1 < −10 is necessary.
This problem becomes more difficult if we are dealing with functions
that change from increasing to decreasing many times, perhaps in an
irregular fashion and perhaps far away from the origin. For this reason,
it is worth noting that if f is a polynomial function of degree n, then it
cannot change directions more than n − 1 times. If we found all n − 1
direction changes, then we can be sure that we did not miss any of
them. We will return to this topic in a later chapter, when we discuss
the derivative of a function.
The preceding example showed why selecting a viewing window
that is too small can be misleading. The next example shows why a
viewing window that is too large can also mislead us. Plot the graph
of the function g(x) = 4x3 + 9x2 + 6x + 1. Using the default viewing
window [−10, 10], or some window containing that one, many software
packages will show a graph that increases everywhere and disappears in
a small interval to the left of 0. This should raise our suspicion that the
program does not properly display the graph of g around 0. Indeed,
g is defined for all real numbers, so its graph should not disappear
anywhere. Taking a closer look, that is, changing the viewing window
to [−1, 1], we see a function that is actually decreasing between x = −1
and x = −1/2.
Trigonometric functions, with their periodicity, are particularly
good examples to demonstrate what software package can and cannot
do. The reader is encouraged to plot the graph of the functions sin x,
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1. FUNCTIONS
cos 2x, tan(x/4), and, finally, sin(1/x) and explain the obtained graphs.
In particular, the reader should try to explain why, for sin(1/x), the
choice of the viewing window is not important as long as it contains
x = 0.
5. Inverse Functions
The inverse f −1 of a function f : A → B “undoes” what f did.
That is, if f (x) = y, then f −1 (y) = x, so f sends x to y, while f 1
sends y back to x. It goes without saying that this f −1 will only be
a function if f −1 (y) is unambiguous, that is, when there is only one
x ∈ A so that f (x) = y. In that case, and only in that case, it is clear
that f −1 (y) = x.
Let us now formalize these concepts.
Definition 1. A function f : A → B is called one-to-one if it sends
different elements into different elements, that is, if x = x implies that
f (x) = f (x ).
One-to-one functions are also called injective functions or injections.
Visually, no horizontal line can intersect the graph of a one-to-one
function more than once.
For instance, if A and B are both the set of real numbers, then
f (x) = x and g(x) = x3 are both one-to-one, but h(x) = x2 is not.
Definition 2. Let f be a one-to-one function with domain A and
range B. Then the inverse of f is the function f −1 : B → A given by
f −1 (y) = x if f (x) = y.
Example 2. Let A and B both be the set of all real numbers. Let
f : A → B be given by f (x) = 2x + 7. Then f −1 (y) = (y − 7)/2.
Solution: If f (x) = y, then y = 2x + 7, so y − 7 = 2x and
(y − 7)/2 = x. As x = f −1 (y), it follows that f −1 (y) = (y − 7)/2.
2
The preceding example shows a general strategy for finding the inverse of a function. Write the equation f (x) = y, with the appropriate
algebraic expression replacing f (x). Then solve for x. If there is more
than one solution, then f is not one-to-one, and so it has no inverse
function. If there is one solution, then that expression is the value of
f −1 (y).
Example 3. If A is the set of positive real numbers, B is the set
of real numbers that are larger
√ than 1, and f : A → B is given by
2
−1
f (x) = x + 1, then f (y) = y − 1.