1549901369-Elements_of_Real_Analysis__Denlinger_

624 Appendix B 11 Sets and Functions

GRAPHS OF FUNCTIONS f : JR--+ JR

Definition B.2.14 The graph of a function f : A __, B , where A, B <:;::: JR, is
the set of all points (x, y) in the Cartesian (rectangular) coordinate system for
which y = f(x ). That is,

graph(!)= {(x, f(x): x E V(f)}.

Thus, a function f : A __, B , where A , B <:;::: JR,

• must pass the vertical line test:
  (a) no vertical line may intersect its graph in more than one point;
  (b) every vertical line that intersects the set A on the x-axis also intersects
  its graph.


• is 1-1 iff it p asses the horizontal line test: no horizontal line may
  intersect its graph at more than one point.


• is onto B iff every horizontal line t hat intersects B on the y-axis also
  intersects its graph.

Vertical

line

3

y = x^2 , not 1-1,

not onto IR

x

Figure B.6

y

EXERCISE SET B.2

y =f(x) =x^3

is 1-1

is onto IR

x

l. For each of the following functions f , find (the largest possible subsets of

JR that could be) V(f) and 'R(f), and tell whether or not f is 1-1 and/or

onto R

(a) f ( x) = 2x - 3

(c) f(x ) = J3x - 4

(e) f(x ) = El

x

(b) f(x) = lxl - 2

( d) f ( x ) = x^2 + 2x + 4

1

( f ) f ( x) = x 2 + 1