Section 1.1 Functions
A relation is a correspondence between two sets. Relations can be described in several
different ways. For example: if x and y are two elements in these sets and if a relation
exists between x and y, they we say that x corresponds to y or that y depends on x and
we denote it as
 An arrow diagram: x  y
 An ordered pair: x, y 
 An equation: for example y  x 2  5 x  6
A function is a special type of relation in which each element from the first set, say X,
corresponds to exactly one element from the second set, say Y. The set X is called the
domain of the function and all elements in Y that are linked to an element in X are
collectively called the range of the function.
Determining When a Relation Represents a Function
For an arrow diagram, a relation is a function if each element in the domain has only one
arrow pointing to an element in the range.
For a set of ordered pairs, check the first element in each ordered pair. If its value is
repeated and their second elements are different, then the relation is not a function.
For an equation, first graph the equation. If a vertical line can be drawn through the graph
of the equation which intersects the graph in more than one location, then the relation is
not a function.
Function Notation
When a relation is in equation form and it has been determined that it is also a function,
we can denote this determination by using function notation. For example, if the relation
is of the form y  x 2  5 x  6 , which is also a function, then a corresponding function
notation is f x   x 2  5 x  6 . In other words, we replace the y with f x  , which reads “
f of x” or “f evaluated at x” and is the value of f at the number x.
Finding the Domain of a Function in Equation Form
The domain of a function in equation form is usually a subset of the real numbers. The
following guidelines can be used to find the domain of a function for certain types of
functions.
 Polynomial functions: The domain of a polynomial function is the set of all real
numbers.
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

Rational functions: The domain of a rational function is the set of all real numbers
except for any real numbers which make the denominator equal to zero, if any.
Radical functions: The domain of a radical function is the set of all real numbers
except for every real number which make the algebraic expression under the
radical less than zero.
Algebra of Functions
 Addition:  f  g x  f x  g x
 Subtraction:  f  g x  f x  g x
 Multiplication:  f  g x  f x  g x
f 
f x 
 Division:  x  
, for g x  0
g x 
g
Difference Quotient
The difference quotient of a function f is the function
f x  h   f x 
, for h  0 .
h
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Section 1.2 The Graph of a Function
When viewing the graph of a function, we can determine many features of the function.
For example, we can determine the function’s
 Values for given inputs (i.e. values of y for given values of x)
 Inputs for given values (i.e. values of x for given values of y)
 Intercepts
 Domain
 Range
just to name a few.
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Section 1.3 Properties of Functions
Increasing, Decreasing, and Constant Functions
A function is said to be increasing on an open interval I if the function values increase as
the values for the independent variable increase over I.
A function is said to be decreasing on an open interval I if the function values decrease
as the values for the independent variable increase over I.
A function is said to be constant on an open interval I if the function values do not
change as the values for the independent variable increase over I.
To determine an increasing, decreasing, or constant function from its graph, view the
graph from left to right.
 If the graph is rising, the function values are increasing.
 If the graph is falling, the function values are decreasing.
 If the graph is neither rising nor falling (i.e. horizontal), the function values are
constant.
Local Minima and Maxima
A function value is called a local maximum (plural, maxima) if it is larger than the
function values of its neighboring points (seen as a peak in the graph).
A function value is called a local minimum (plural, minima) if it is smaller than the
function values of its neighboring points (seen as a valley in the graph).
A function may have more than one local maximum or local minimum.
Even and Odd Functions
A function that is symmetric with respect to the y-axis is also called an even function. In
other words, a function is even if for every number x in its domain the number – x is also
in the domain and f  x  f x .
A function that is symmetric with respect to the origin is also called and odd function. In
other words, a function is odd if for every number x in its domain the number – x is also
in the domain and f  x   f x .
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Section 1.4 Library of Functions; Piecewise-defined Functions
We have encountered many different types of functions. We need to know the
fundamental shapes of their graphs, as well as a few functions we have not seen yet, so
that we can recognize whether a graph is in a particular class of functions. It will also
help us to graph functions that are in the same class of functions.
We will consider the following sets of functions.
 Types of Linear Functions
o General form: f x  mx  b
o Constant function: f x  b
o Identity function: f x   x
 Square Function f x   x 2
 Cube Function f x   x 3






Square Root Function f  x   x
Cube Root Function f  x   3 x
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Reciprocal Function f x  
x
Absolute Value Function f x   x
Greatest Integer Function f x  int x , which is the greatest integer less than
or equal to x.
Piecewise Defined Functions are functions that are defined differently on
different parts of its domain.
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Section 1.5 Graphing Techniques: Transformations
Shifts and Scaling of Functions
We can shift, scale, or reflect any function, f (x), as follows:
Shifts:
 For a  0 ,
 f ( x)  a
 f ( x)  a
 f ( x  a)
 f ( x  a)
produces a vertical shift upwards a units.
produces a vertical shift downwards a units.
produces a horizontal shift to the right a units
produces a horizontal shift to the left a units
Scaling and Reflections:
 For a  f (x)
 If a  1 , the graph is stretched vertically
 If 0  a  1, the graph is compressed vertically
 If a  0 , the graph is reflected across the x-axis
 For f (ax )
 If a  1 , the graph is compressed horizontally
 If 0  a  1, the graph is stretched horizontally
 If a  0 , the graph is reflected across the y-axis
We can use these facts along with the basic function shapes we learned in Section 1.4 to
graph other functions in the same class that were “transformed” in at least one way.
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Section F.1 The Distance and Midpoint Formulas
Distance Between Two Points and the Midpoint Formula
The distance between two points P1  x1 , y1  and P2  x 2 , y 2  , denoted by d P1 , P2  ,
is d P1 , P2  
x2  x1 2   y 2  y1 2 .
The distance formula above can be used to verify whether a triangle is a right triangle. To
do this, find the squares of the lengths of all three sides of the triangle. If the squares of
two lengths add up to the square of the remaining side, it is a right triangle.
The midpoint M  x, y  of the line segment from P1  x1 , y1  to P2  x 2 , y 2  is
 x  x 2 y1  y 2 
M   x, y    1
,
.
2 
 2
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Section F.4 Circles
Circles and Their Equations
A circle is a set of points in the xy-plane that are a fixed distance r from a fixed point
h, k  . The fixed distance r is called the radius of the circle, and the fixed point h, k  is
called the center of the circle.
The standard form of an equation of a circle with radius r and center h, k  is
 x  h 2   y  k 2  r 2 .
The general form of an equation of a circle is x 2  y 2  ax  by  c  0 .
If we are given the standard form we can find the general form by simplifying the
standard form and setting it equal to zero.
If we are given the general form we can find the standard form by completing the square
on the x terms and the y terms in the general form, placing all remaining constants on the
right hand side of the equation.
Completing the Square
Recall that a perfect square trinomial is of the form
a  b 2  a 2  2ab  b 2
a  b 2  a 2  2ab  b 2
Unfortunately, not all quadratic expressions are perfect square trinomials. However, we
can rewrite them as a perfect square trinomial by a process called completing the
square.
2
b
Given a quadratic expression of the form x 2  bx , if we add the constant term   to
2
2
2
b
b 
the expression it results in the expression x 2  bx      x   , which is a perfect
2
2 
square trinomial.
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