Name
September 3, 2014
Math 4 class notes and problems
section 1.2 page 1
Functions, domain and range
in textbook: pp. 80-81
NOTES:
Important basic question types about functions
This is a follow-up to our last class work.
question
algebraic meaning
“Find f(2).”
For input x = 2, what is the
output?
“Find the zeros of
f(x).” or
“Solve f(x) = 0”.
“Solve f(x) = 2.”
What input value x results in
an output value of 0?
What input value x results in
an output value of 2?
“Solve f(x) = g(x).” For what input value x do
functions f and g have equal
outputs?
graphical meaning
For the point with x-coordinate 2, what is
the y-coordinate?
Calculator: [2nd][CALC] value.
What are the x-intercept(s) of the graph?
Calculator: [2nd][CALC] zero.
For the point with y-coordinate 2, what are
the x-coordinate(s)?
Calculator: Graph Y1 = f(x), Y2 = 2, then
use [2nd][CALC] intersect.
What are the x-coordinates of the
intersection points of the f and g graphs?
Calculator: [2nd][CALC] intersect.
Definitions: domain and range
in textbook: pp. 80-81
The domain is the set of possible input values (or x-values or x-coordinates) for a function.
The range is the set of possible output values (or y-values of y-coordinates) for a function.
Interval notation
in textbook: p. 4
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 [ ] and/or ( ) brackets. Here are some examples.
“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 7 < y < ∞.”
“The range is (7, ∞).”
∞ 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
Notations that aren’t specific enough: “1 to 5”, “1  5”, “1 … 5”, “7 and up”, “7 ,” etc.
Name
September 3, 2014
Math 4 class notes and problems
section 1.2 page 2
Exercises: graphical
1. State the domain and range for each function graphed below
a.
b.
2. Sketch a function graph that fits each of the following.
a. Domain is [2, 5] and range is [–3, 4].
b. Domain is 0 < x ≤ 1 and range is 2 ≤ y < ∞.
Name
September 3, 2014
Math 4 class notes and problems
section 1.2 page 3
Exercises: algebraic
When you’re given a function formula and asked to identify the domain, that means that you
need to describe all the real numbers x that are possible as inputs to the formula. (The main
algebraic reasons that an input would be impossible are: resulting in division by zero; resulting in
square root of a negative number). See page 82 Example 3 for examples.
3. Identify the domain of each of these functions.
a.
f x  
x 1
x2
b. f x   x 2  4
c. f x  
x
x 1
2
4. Identify the domain and the range of each function. It may help to make the graph first,
either by hand or by calculator.
a.
f x   x 2  2 x  5 , for –2 ≤ x < 4
b.
f ( x) 
1
, for all real numbers x that are possible as inputs
x3
Exercises: numerical
5. 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.
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?
x
–4
–2
0
2
4
f(x)
9
3
1
3
9
Exercises: in context
For functions in geometric contexts and real-world contexts, the domain and range are limited by
which values would make sense in the context. Remember the open box problem and see p. 82
Example 3c.
6. Find the domain and range of each function.
a. function A(n) = (n – 2)  180 which gives the angle sum of an n-sided polygon
b. function T(x) = 0.0625 x which gives the Massachusetts sales tax on an $x purchase
Name
September 3, 2014
Math 4 class notes and problems
section 1.2 page 4
Optional challenge problems: if you are feeling good about problems 1-6,
try these:
7.
8.
9. More practice with domain and range:
a. Write the equation of a linear function, in point-slope form, that goes through the
points (2,‒3) and (5,4).
b. Make up a function whose domain is [2, 5] and range is [–3, 4]. You should specify
the x values for which the formula applies (see exercise 7, for example).
c. Make up a function whose domain is [2,5) and range is [–3, 4).
d. Make up a function whose domain is [2, 5) and range is (–3, 4]. Careful, this one is
harder.
e. Challenge: Make up a function whose domain is 0 < x ≤ 1 and range is 2 ≤ y < ∞.
Homework for Wednesday night

Finish problems 1–6 from this packet. Do problems 7-9 (optional).