Notation for End Behavior of functions
In our discussion in class about the end behavior of a function these are the key points presented:
1. The end behavior means what happens to the values of the function, heights, when the values in the domain
go toward that end point
2. Some times the values of the heights are not obtained (we use " " to represent them), or sometimes they are
attained.
The examples below will give you a good illustration of our notation. The functions in those graphs correspond to fHxL = x3 , gHxL = 3x , hHxL =
x
Domain H-¥, ¥L
Lim fHxL = "-¥"
x® -¥
Domain H-¥, ¥L
Lim gHxL = "0- "
x® -¥
Lim fHxL = "¥"
x® ¥
Lim gHxL = "¥"
x® ¥
Domain : x ³ 0
Lim+ hHxL = 0+ Hzero is attainedL
x® 0
Lim fHxL = "¥"
x® ¥
2
End Behavior Notes.nb
COMPARING FUNCTIONS
Functions can be compared only on their common domain. Below you find pairs of functions. In front of each
pair you find their common domain. @0, ¥L
fHxL = x2 , gHxL = x3
hHxL = x3 , rHxL =
yHxL = x, tHxL =
1
x
x
H-¥, ¥L
H-¥, 0L Ü H0, ¥L
@0, ¥L
When solving inequalities Hcomparing functionsL such as hHxL > rHxL keep in mind these rules :
1. Functions can be compared only on their common domain
2. Need to find the values where the expressions are equal (in our example x=1). Those values determine
the subintervals where the inequality is tested.
3. Pick test points on each subinterval.
4. Check whether the test points satisfy the inequality. When the test point checks, then any point on that
subinterval checks.
In the case of h(x) and r(x), the common domain is (-¥, 0) Ü (0, ¥) and their intersection point is at x=1. So
the number line with the subintervals to check where the inequality is true is
0
1