1.2 Functions and Their Properties
A function is actually a simple concept; if it were not, history would have replaced it with a
simpler one by now! Here is the definition:
Definition: Function, Domain, and Range
A function from a set D to a set R is a rule that assigns to every
element in D a unique element in R. The set D of all input
values is the __________ of the function, and the set R of all
output values is the ____________ of the function.
There are many ways to look at functions. At some point in your math career, your teacher may
have talked to you about a “function machine”, in which values of the domain (x) are fed into the
machine (the function f) to produce range values (y). A very concise way to state this is using the
function notation: y = f(x), read as “y equals f of x”.
A function can also be viewed as a mapping of the elements of the domain onto the elements of
the range. We’ll sketch two figures below: one that is a function and one that is not.
FUNCTION MAPPING:
NOT A FUNCTION MAPPING:
The uniqueness of the range value is very important. Knowing that f(2) = 8 tells us something
about f, and that understanding would be contradicted if we were to discover later that f(2) = 4.
Why will you never see a function defined by an ambiguous formula like f(x) =3𝑥 ± 2?
Another useful way to look at functions is graphically. The graph of the function y = f(x) is the
set of all points (x, f(x)), with x in the domain of f. We match domain values along the x-axis to
get the ordered pairs that yield the graph of y = f(x). Draw a graph that is a function and a graph
that is not a function
FUNCTION SKETCH:
NOT A FUNCTION SKETCH:
One useful way to determine whether or not a relation is a function graphically is to use the
vertical line test.
Vertical Line Test
A graph (set of points (x, y)) in the xy-plane defines y as a
function of x if and only if no vertical line intersects the graph in
_____________________________.
Function or not?
Now that we know what a function is and what one does and does not look like we need to think
about other terminology that can be used to describe a function. While some of the terms that
follow may be familiar to you, we will take time to carefully review each one by giving a formal
definition and then reviewing some illustrative examples of that concept to help cement your
understanding.
Domain and Range
Domain
The inputs where the function is defined.
Exclude x-values that divide by zero or take an
even root of a negative number.
The output values of the function.
Range
Generally, we will define functions algebraically and we will assume that the domain of the
function defined by the algebraic expression is the same as the domain of the algebraic
expression itself. That is, we will use the implied domain. Here is an example that should help
illustrate this point: The volume formula for a circle is: 𝐴 =
4
3
𝜋𝑟 3 and the implied domain for
this formula would be all real numbers; however, the volume function is not defined for negative
radius values. This means that if our intent were to study the volume function, we would need to
restrict the radius to values of 0 or higher (we call this the relevant domain).
Example 1: Finding the Domain and Range of a Function
Find the domain and range of each of these functions:
(a) 𝑓(𝑥) = √𝑥 + 3
(b) 𝑔(𝑥) =
√𝑥
𝑥−5
(c) ℎ(𝑥) =
√3 2
𝑥
4
While a formal definition of continuity involving limit notation is vital to the study of Calculus,
an informal understanding will be sufficient at this time. Basically, the mathematical definition
of continuity captures two ideas: (1) the values of a function f(x) at points near ‘a’ are good
predictors of the value of f at ‘a’ and (2) the graph of f is a connected curve with no jumps, gaps,
or holes. If either of these conditions is violated, the function is considered to be discontinuous.
There are two types of discontinuities to consider – removable and non-removable.
A removable discontinuity is one at which the limit of the function exists, but it does not equal
the value of the function at that point. Informally, we think of this as a discontinuity that can be
“repaired” – it is a point at which a graph is not connected but can be made connected by filling
in a single point.
A non-removable (or essential) discontinuity is one at which the limit of the function does not
exist. Informally, it is a place where a graph is not connected and cannot be made connected
simply by filling in a single point. Step (or jump) discontinuities and vertical asymptotes (which
create an infinite discontinuity) are two types of non-removable discontinuities.
Let’s illustrate these ideas with a few graphs:
Continuous
Continuity
Removable
Removable
(Hole, f(a) exists)
(Hole, f(a) does not exist)
Non-removable (Step
Non-removable
Non-removable
or Jump)
(Infinite, vertical asymptote
on one side)
(Infinite, vertical asymptote
on both sides)
Example 2: Determining Continuity
Determine if each function is continuous. If the function is not continuous, classify the type of
discontinuity (removable or non-removable) and its location.
𝑥2
(a) 𝑓(𝑥) = − 2𝑥+4
(b) 𝑓(𝑥) =
(c) 𝑓(𝑥) =
𝑥+1
𝑥 2 −𝑥−2
𝑥+1
𝑥 2 +𝑥+1
(d) 𝑓(𝑥) = {
−𝑥 2 , 𝑥 ≠ 1
0, 𝑥 = 1
A function
𝑓(𝑥 )𝑖𝑠 …
Increasing
Constant
Decreasing
Increasing and Decreasing
If for any 𝑥1 , 𝑥2 In less formal terms:
on (a, b) with
𝑥1 < 𝑥2
𝑓(𝑥1 ) < 𝑓(𝑥2 ) GOING UP:
a positive change in x results in a
positive change in f(x)
𝑓(𝑥1 ) = 𝑓(𝑥2 ) ALWAYS THE SAME:
a positive change in x results in
zero change in f(x)
𝑓(𝑥1 ) > 𝑓(𝑥2 ) GOING DOWN: a positive change
in x results in a negative change in
f(x)
Example 3: Analyzing a Function for Increasing-Decreasing Behavior
For each function, tell the intervals on which it is increasing and the intervals on which it is
decreasing.
(a) 𝑓(𝑥) = (𝑥 + 2)2
(b) 𝑔(𝑥) =
𝑥2
𝑥 2 −1
The next term that can be used to describe function behavior is boundedness.
Boundedness
A function f is…
…bounded below
If there is some number b, such that b is
____________________________________.
Any such number b is called a lower bound of f.
…bounded above
If there is some number B, such that B is
____________________________________.
Any such number B is called an upper bound of f.
…bounded
If it is bounded both above and below.
…not bounded
If it is not bounded above or below.
Example 4: Determining Boundedness
Identify each of these functions as bounded below, bounded above, bounded or not bounded.
(a) 𝑓(𝑥) = −𝑥 2
(b) 𝑓(𝑥) = 3𝑥 3 − 7
(c) 𝑓(𝑥) = 3𝑥 2 − 4
(d) 𝑓(𝑥) =
𝑥
1+𝑥 2
Many graphs are characterized by peaks and valleys where they change from increasing to
decreasing and vice versa. These points are called extrema.
Local
Maximum
Extrema
A value f(c) that is _______________________
all range values of f on some open interval
containing c.
Local extrema are also called relative extrema.
Absolute
Maximum
A value f(c) that is _______________________
all range values of f.
Local
Minimum
A value f(c) that is _______________________
all range values of f on some open interval
containing c.
Local extrema are also called relative extrema.
Absolute
Minimum
A value f(c) that is _______________________
all range values of f.
Example 5: Determining Local Extrema
Determine whether 𝑓(𝑥) = 𝑥 4 − 7𝑥 2 + 6𝑥 has any local maxima or local minima. If so, find
each local maximum or minimum value and the value of x at which each occurs and then
determine if any of these values are absolute extrema.
There are two particular types of symmetry that we might look for in a function: symmetry with
respect to the y-axis and symmetry with respect to the origin. Generally we are not concerned
with symmetry with respect to the x-axis, since graphs with this type of symmetry are not
function (with the exception of y = 0).
Symmetry
Graph
…with respect to
the y-axis
Algebra
𝑓 (𝑥 ) = 𝑓(−𝑥)
Functions with this property
are called even functions.
…with respect to
the origin
𝑓(−𝑥 ) = −𝑓(𝑥)
Functions with this property
are called odd functions.
Example 6: Determining Symmetry
Tell whether each of the following functions is odd, even, or neither.
(a) 𝑓(𝑥) = 𝑥 2 − 3
(b) 𝑔(𝑥) = 𝑥 2 − 2𝑥 − 2
(c) ℎ(𝑥) =
𝑥3
4−𝑥 2
Asymptotes
A graph may approach a horizontal or vertical line but never quite reach it. It may also approach
𝑥3
a curve (for example, 𝑥+1 gets awfully close to 𝑥 2 for large values of x), but the formal definition
of an asymptote is again more rigorous than what we require here.
Horizontal asymptotes can be found by simply looking at your graph and seeing if it flattens off
towards a certain value as x tends toward positive and negative infinity.
For rational functions,



when the degree of the numerator and the denominator are the same, the horizontal
asymptote is found by dividing the coefficients of the leading terms.
when the degree of the denominator is larger than the degree of the numerator, the
horizontal asymptote is the x-axis (y = 0), and
when the degree of the denominator is smaller than the degree of the numerator, there is
no horizontal asymptote, as there is instead an oblique or slant asymptote.
Vertical asymptotes will most often correspond to the zeroes of the denominator of a rational
function.
Example 7: Identifying Asymptotes
Identify any horizontal or vertical asymptotes for the following functions:
(a) 𝑓(𝑥) =
𝑥−1
𝑥
2𝑥−4
(b) 𝑔(𝑥) = 𝑥 2 −4
(c) ℎ(𝑥) =
𝑥
𝑥 2 −𝑥−2