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0.1 Functions and Their Graphs
Real Numbers
• A set is a collection of objects.
• The real numbers represent the set of numbers
that can be represented as decimals.
• We distinguish two different types of decimal
numbers
o A rational number is a decimal number that may can
be written as a finite or infinite repeating decimal, such
as  2 or = 2.333…
Real Numbers (2)
An irrational number is a decimal number that
does not repeat and does not terminate,
such as = -1.414214… and  = 3.14159…
• The real number set is represented
geometrically by the real number line
Real Numbers (3)
• We use inequalities to compare real
numbers
x y
x is less than y
x y
x is less than or equal to y
x is greater than y
xy
x is greater than or equal to y
x y
Real Numbers (4)
• We often use a double inequality such as
abc
as shorthand for a pair of inequalities
ab
and
bc
When we use double inequalities, the positions of
a, b, and c must be written as they would appear
on the real number line if read from right to left or
left to right.
Real Numbers (5)
• Inequalities can be expressed geometrically
or by using interval notation.
Real Numbers (6)
Real Numbers (7)
• The symbols

and

do not represent actual numbers, but
indicate that the corresponding line
segments extend infinitely far to the left or
right.
Functions
• A function of a variable x is a rule f that assigns to
each value of x a unique number f(x) (read ”f of
x”), called the value of the function at x.
• x is called the independent variable.
• The set of values that the independent variable is
allowed to assume is called the domain of the
function. A function’s domain might be explicitly
specified as part of its definition or might be
understood from its context.
• The range of a function is the set of values that
the function assumes.
Functions (2)
• Examples of functions:
f(x) = 3x –1
f(x) = 9x3 + 7x2 – 8
Functions (3)
• Function Example: If we let f be the function with
the domain of all real numbers x, and defined by
the formula f(x) = 4x2 – 2x + 6, we can find the
corresponding value in the range of f for a given
value of x by substitution.
Find f(2): f(2) = 4(2)2 – 2(2) + 6 = 4(4) – 4 + 6 =
16 – 4 + 6 = 18
Find f(-3): f(-3) = 4(-3)2 – 2(-3) + 6 = 4(9) + 6 + 6
= 36 + 6 + 6 = 48
Functions (4)
• Function Example: Let f be the function
with the domain of all real numbers x, and
defined by the formula
f(x) = (4 – x)/(x2 + 7).
4h
Find f(h): f (h)  2
h 7
Find f(h+2): f(h+2) =
4  (h  2)
2
(h  2)  7
Functions (5)
• When the domain of a function is not explicitly
specified, it is understood that the domain of the
function is all values for which the defining
formula makes sense.
The domain of f for f(x) = 2x is all real numbers.
The domain of f for f(x) = 1/x is all real numbers
except x = 0.
The domain of f for f ( x)  x is all real numbers
greater than or equal to 0.
Graphs of Functions
• We can express a function geometrically by
expressing it as a graph in a rectangular xycoordinate system.
• Given any x in the domain of a function f, we can
plot the point (x, f(x)). This is the point in the xyplane whose y-coordinate is the value of the
function at x.
• It is possible to approximate the graph of the
function f(x) by plotting the points (x,f(x)) for a
representative set of values of x and joining them
by a smooth curve.
• We often use a tabular construct to capture the
points we wish to plot.
Graphs of Functions (2)
Graphs of Functions (3)
Graphs of Functions (4)
• To every x in its domain, a function assigns one
and only one value of y. That value is precisely
f(x). This means:
The variable y is called the dependent variable
because its value depends on the value of the
independent variable x.
– For every curve that is the graph of a function,
there is a unique y such that (x,y) is a point on
the curve.
Not every geometric curve is the graph of a
function.
Graphs of Functions (5)
Graphs of Functions (6)
• Vertical Line Test – A curve in the xy-plane
is the graph of a function if and only if no
vertical line can be drawn that will intersect
the curve at more than one point.
• Which curves are the graphs of functions?
Three Views of a Function
• We how have three ways to describe a function
• By giving the formula f(x) = … and defining the
domain of the independent variable (x). A
function specified in terms of a formula is said to
be defined analytically.
• By drawing the functions graph. Such a function
is said to be defined graphically.
• By providing a table of function values (x and
f(x)). This method is said to describe a function
numerically.
Graphs of Functions (7)
Graphs of Equations
• The equations arising in connection with functions
all have the form:
y = [an expression in x]
• Note that not all equations connecting x and y are
functions.
• A graph of an equation can be plotted the same as
the graph of a function. However, the resulting
graph will only pass the Vertical Line Test if the
equation represents a function.
Graphs of Equations (2)