Page 46 - Calculus Demystified

Basics
CHAPTER 1
You Try It: Does the expression 33

4 if x< 3
g(x) = 2
x − 7if x ≥ 2
define a function? Why or why not?
EXAMPLE 1.27
√
Let f(x) =± x. Discuss whether f isa function.
SOLUTION
This f can only make sense for x ≥ 0. But even then f is not a function
since it is ambiguous. For instance, it assigns to x = 1 both the numbers 1
and −1.

1.8.2 GRAPHS OF FUNCTIONS
It is useful to be able to draw pictures which represent functions. These pictures,
or graphs, are a device for helping us to think about functions. In this book we will
only graph functions whose domains and ranges are subsets of the real numbers.
We graph functions in the x-y plane. The elements of the domain of a function
are thought of as points of the x-axis. The values of a function are measured on the
y-axis. The graph of f associates to x the unique y value that the function f assigns
to x. In other words, a point (x, y) lies on the graph of f if and only if y = f(x).

EXAMPLE 1.28
2
Let f(x) = (x + 2)/(x − 1). Determine whether there are pointsof the
graph of f corresponding to x = 3, 4, and 1.

SOLUTION
The y value corresponding to x = 3is y = f(3) = 11/2. Therefore the
point (3, 11/2) lies on the graph of f . Similarly, f(4) = 6 so that (4, 6) lies on
the graph. However, f is undefined at x = 1, so there is no point on the graph
with x coordinate 1. The sketch in Fig. 1.38 was obtained by plotting several
points.

Math Note: Notice that for each x in the domain of the function there is one and
only one point on the graph—namely the unique point with y value equal to f(x).
If x is not in the domain of f , then there is no point on the graph that corresponds
to x.
EXAMPLE 1.29
Isthe curve in Fig. 1.39 the graph of a function?