DrDelMath
SUMMARY
Chapter 2: Functions
and their Graphs
Section 2.1: Linear Equations in Two Variables
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See the Supplemental Material. It contains everything about linear equations.
Section 2.2: Functions
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Definition:
The
symbol f(x) is read: f of x
Note that the element inside the parenthesis is
a single element of the domain
Note that f(x) is a single element of the range
of the function.
In
reference to the notation above, it is correct
to speak of the function f.
In reference to the notation above it is incorrect
to speak of the function f(x).
Functional notation is most commonly used to state the rule of a function. This use of functional notation is especially convenient when the rule for the function can be expressed in a closed form in terms of the domain element.
Convention: Common practice is to write only the rule for a function with no mention of the domain or range. To avoid confusion the following convention has been adopted. Unless otherwise stated, the domain of a function is the largest set of real numbers for which the rule make sense (has meaning) and the range is the set of real numbers associated with those domain elements.
Definition: The coordinates of a point (a, b) are said to satisfy the rule of a function f if b = f(a).
Definition: The graph of a function is the set of all points whose coordinates satisfy the rule of the function.
Equivalent Definition: The previous two definitions imply that the graph of a function is the set of all points whose coordinates are (a, f(a)) where a is an element of the domain and f(a) is the corresponding range element.
Definition: A zero of a function f is a domain element a for which f(a) = 0.
Definition: A function f is called a constant function if there is some real number k such that f(a) = k for all a in the domain of f.
Definition: The identity function is the function I which has the property that I(a) = a for all a in the domain of I.
Definition: The zero function z is the function defined by z(a) = 0 for all a in the domain of z.
Definition: The sum of two functions f and g is a function h with the property that h(a) = f(a) + g(a) for all a in the common domain of f and g. The function h is usually symbolized by f + g, so that we have (f + g)(a) = f(a) + g(a).
Definition: The additive inverse of a function f is a function h with the property that h + f = z.
Definition: The product of two functions f and g is a function h such that h(a) = f(a)g(a) for all a in the common domain of f and g. The function h is usually symbolized by fg, so we have fg(a) = f(a)g(a).
Definition: The multiplicative inverse of a function f is a function h with the property that hf = I
Definition: A linear function is a function whose rule may be written in the form f(x) = mx + b where m and b are real numbers.
Definition: A quadratic function is a function whose rule may be written in the form f(x) = ax2 + bx + c where a, b, and c are real numbers and a is not 0.
Definition:
A polynomial
function is a function whose rule may be written in the form
f(x) = anxn + an-1xn-1 + ... + a1x
+ a0 where each ai is a real number and n is a natural
number.
Definition: A rational function is a function whose rule may be written as the quotient of two polynomial functions.
Section 2.3: Analyzing Graphs of Functions
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Definition: The coordinates of a point (a, b) are said to satisfy the rule of a function f if b = f(a).
Definition: The graph of a function is the set of all points whose coordinates satisfy the rule of the function.
Definition: A zero of a function f is a domain element a for which f(a) = 0.
Vertical Line Test: If a vertical line may be drawn so that it intersects a graph in more than one point, then that graph is not the graph of a function.
Definition: A function f is increasing on an interval if, for any x1and x2 in the interval, x1< x2 implies f(x1) < f(x2)
Definition: A function f is decreasing on an interval if, for any x1 and x2 in the interval, x1< x2 implies f(x1) > f(x2)
Definition: A function f is constant on an interval if, for any x1 and x2 in the interval f(x1) = f(x2)
Definition: A function f is even if for each x in the domain of f, f(x) = f(-x)
Definition: A function f is odd if for each x in the domain of f, - f(x) = f(-x)
Section 2.4: A Library of Functions
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Definition: A linear function is a function whose rule may be written in the form f(x) = mx + b where m and b are real numbers.
Definition: A quadratic function is a function whose rule may be written in the form f(x) = ax2 + bx + c where a, b, and c are real numbers and a is not 0.
Definition: A function f is called a constant function if there is some real number k such that f(a) = k for all a in the domain of f.
Definition: The absolute value function is a function whose rule may be written in the form f(x) = | x |
Definition: The zero function z is the function defined by z(a) = 0 for all a in the domain of z.
Definition:
The reciprocal
function is a function whose rule may be written in the form
Definition: The identity function is the function I which has the property that I(a) = a for all a in the domain of I.
Definition: The squaring function is the quadratic function f whose rule may be written in the form f(x) = x2.
Definition: The cubing function is the cubic function f whose rule may be written in the form f(x) = x3.
Definition:
The square
root function is the function f whose rule may be written in
the form 
Definition:
The greatest
integer function is the f whose rule may be written in the form
Section 2.5: Shifting, Reflecting, and Streching Graphs
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Section 2.6: Combinations (Arithmetic) of Functions
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Definition: The identity function is the function I which has the property that I(a) = a for all a in the domain of I.
Definition: The zero function z is the function defined by z(a) = 0 for all a in the domain of z.
Definition: The sum of two functions f and g is a function h with the property that h(a) = f(a) + g(a) for all a in the common domain of f and g. The function h is usually symbolized by f + g, so that we have (f + g)(a) = f(a) + g(a).
Definition: The additive inverse of a function f is a function h with the property that h + f = z.
Definition: The difference of a function f minus a function g is a function h with the property that h(a) = f(a) - g(a) for all a in the common domain of f and g. The function h is usually symbolized by f - g, so that we have (f - g)(a) = f(a) - g(a).
Definition: The product of two functions f and g is a function h such that h(a) = f(a)g(a) for all a in the common domain of f and g. The function h is usually symbolized by fg, so we have fg(a) = f(a)g(a).
Definition: The multiplicative inverse of a function f is a function h with the property that hf = I
Definition:
The quotient
of a function f divided by a function
g is a function h with the property that h(a) = 
for
all a in the common domain of f and g. The function h is usually symbolized
by 
,
so that we have 
.
Definition:
The composition
of a function f with a function g is a function named 
whose
rule is 
The domain of
is the set of all x in the domain of g for which g(x) is in the domain of f.
The following picture is essential to understanding composition
of functions.
Section 2.7: The Inverse of Functions
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Definition:
The inverse
of a function f.
Let f be a function with domain A and range B. Then the inverse of the function,
if it exists, is a function named 
,
with domain B and range A with the property that
Note that the inverse of a function is the inverse with respect to composition.
From the symmetrical nature of the above definition it is clear that
is the inverse of f and conversely, f is the inverse of .
Fact:
The
functions f and
are inverses of each other.
The symbol
is read as "f inverse".
The following diagram illustrates the relationship between a function and it
inverse.
Fact: Not all functions have inverses.
Fact:
(Horizontal Line Test) If
there is a horizontal line which intersects the graph of a function f in more
than one point, then the function f does not have an inverse.
If no horizontal line intersects the graph of a function f in more than one
point, then the function f has an inverse .
Definition: A function is called a one-to-one function if no element of the range is the associate of more than one domain element.
In terms of our fundamental "arrow" description of a function this means that no range element has more than one arrow ending at it.
The horizontal line test is a test to determine if a function is one-to-one.
Fact:
A
function f has an inverse
if and only if f is a one-to-one function.
Process: The following steps is a method for Finding the Inverse of a Function. Suppose the function is named f.
to obtain the rule for .Note: Replacing f(x) with a single variable y is for convenience only. The entire process can be carried out without such replacement, but the notation becomes awkward.
It is INCORRECT
to speak of an inverse function. Just as it is incorrect to speak of 3/4 as
an inverse.
It is correct to speak of the inverse of a function. Just as it is correct to
speak of 3/4 as the inverse of 4/3.
Thus in the above process we find the inverse of a function.