Name
September 4, 2013
Honors Advanced Math
Homework problem set page 1
Describing functions: domain and range
This homework assignment pertains to some of the material in section 1.2 of the
Demana Precalculus textbook. If you need to review the definitions of the concepts
function, domain, and range, read the beginning of that section. This problem set is due on
Monday September 9.
Interval notation
Stating the domain or range of a function often requires describing an interval of values, with or
without its endpoints. Intervals can be described either using inequalities or using a special
notation involving square and/or curved brackets. These examples show two equivalent ways of
describing the same intervals.
“The range is 1 ≤ y ≤ 5.”
“The range is [1, 5].”
[ ] brackets mean endpoints included
“The domain is 3 < x ≤ 6.”
“The domain is (3, 6].”
( ) brackets mean endpoints excluded
“The range is 2 < y < ∞.”
“The range is (2, ∞).”
∞ used when range extends infinitely
“The domain contains all x such that 2 ≤ x < 4 or 6 < x ≤ 8.”
“The domain is [2, 4)  (6, 8].”
“The domain consists of x = 1, x = 3, and x = 5.”
“The domain is {1, 3, 5}.”
 symbol joins two intervals
{ } used for lists of isolated numbers
Exercises
In exercises 1–2, state the domain and range of the functions whose graphs are shown.
1.
2.
For each f(x) formula given in Exercises 3–5, identify the domain (that is, the largest possible set
of real numbers x that can be used as inputs to the formula), by reasoning about which inputs are
possible and which are not.
Name
September 4, 2013
3.
f x   x 2  4
Honors Advanced Math
Homework problem set page 2
4.
f x  
x 1
x  5x  6
2
5.
f x  
x
( x  1)( x 2  1)
Identify the domains and ranges of the functions defined in exercises 6–9. For some problems it
will help to draw the graph first.
6.
7.
8.
9.
f x   x 2  2 x  5 , for –2 ≤ x < 4.
3 x
, for all real numbers x that are possible as inputs.
f ( x) 
x3
for
x  0;
 1x ,

f x    x  2, for 0  x  3;
 1,
for
x  3.

 x 2  3, for x  0;

f x    4 x  x 2
 x  4 , for x  2.
10. Both parts of this problem refer to the input-output table at the right.
a. If the only points of a function are the five points shown in the
table, state the domain and range of this function.
x
–4
–2
0
2
4
f(x)
9
3
1
3
9
b. Suppose there is a quadratic function (parabola graph) and the
table shows five of its many points. What are the domain and range of this quadratic
function?
In exercises 11–14, make up functions whose domain and range are as stated. It may help to
draw a graph first. Give each answer as an f(x) formula. You may also specify the x values for
which the formula applies (as seen in exercise 6, for example).
11. Domain is [2, 5] and range is [–3, 4].
12. Domain is [2, 5) and range is (–3, 4].
13. Domain is 0 < x ≤ 1 and range is 2 ≤ y < ∞.
14. Domain is [1, 2]  (3, 4) and range is (5, 10)  [20, 30].
Hint for 14: You’ll need to use a “piecewise” f(x) definition (as seen in exercises 8 and 9).
15. Consider function f(x) as defined below. Determine values of m and b such that the range of
this function is [–2, 10]. (There are two answers to this problem; find both.)
for
x  3;
 2,

f x   mx  b, for  3  x  3;
 4,
for
x  3.
