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Transcript Chapter 1

Functions
Representations of Functions
There are four possible ways to represent a function:
•
verbally
(by a description in words)
•
numerically
(by a table of values)
•
visually
(by a graph)
•
algebraically (by an explicit formula)
A
The area A of a circle depends on the radius r of the circle.
The rule that connects A and r is given by the equation:
A r
2
With each positive number r there is associated one value
of A, and we say that A is a function of r
B
C
The rule that the U. S. Postal Service used as of 2001 is as follows:
The cost is 34 cents for up to one ounce, plus 22 cents for each
successive ounce up to 11 ounces.
D
Graphs of Functions
The graph of a function is a curve in the xy-plane. But the
question arises: Which curves in the xy -plane are graphs of
functions? This is answered by the following test.
The Vertical Line Test A curve in the xy-plane is the
graph of a function of x if and only if no vertical line
intersects the curve more than once.
Classification of Functions
We may classify functions by their formula as follows:
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Polynomials
Linear Functions, Quadratic Functions. Cubic Functions.
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Piecewise Defined Functions
Absolute Value Functions, Step Functions
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Rational Functions
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Algebraic Functions
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Trigonometric and Inverse trigonometric Functions
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Exponential Functions
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Logarithmic Functions
Function’s Properties
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We may classify functions by some of their
properties as follows:
Injective (One to One) Functions
Surjective (Onto) Functions
Odd or Even Functions
Periodic Functions
Increasing and Decreasing Functions
Continuous Functions
Differentiable Functions
Symmetry
Transformations of Functions
Combinations of Functions
Composition of Functions
Power Functions
Exponential Functions
Inverse Functions
Logarithmic Functions
The logarithm with base is called the natural logarithm and has a
special notation:
Inverse Trigonometric Functions
When we try to find the inverse trigonometric functions, we have a slight
difficulty. Because the trigonometric functions are not one-to-one, they
don’t have inverse functions. The difficulty is overcome by restricting the
domains of these functions so that hey become one-to-one.