functions. representations of functions there are four possible ways to represent a function:...
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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 of a circle depends on the radius of the circle. The rule that connects and is given by the equation: With each positive number there is associated one value of, and we say that is a function ofTRANSCRIPT
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Functions
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Representations of FunctionsThere 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 of a circle depends on the radius of the circle. The rule that connects and is given by the equation:
With each positive number there is associated one value of , and we say that is a function of
A rA r
2 rA r
A A r
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B
C
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D
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.
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Graphs of Functions The graph of a function is a curve in the -plane.
But the question arises: Which curves in the -plane are graphs of functions? This is answered by the following test.
The Vertical Line Test A curve in the -plane is the graph of a function of if and only if no vertical line intersects the curve more than once.
xyxy
xyx
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Classification of FunctionsWe may classify functions by their formula as follows:• Polynomials Linear Functions, Quadratic Functions. Cubic
Functions.• Piecewise Defined Functions Absolute Value Functions, Step Functions• Rational Functions• Algebraic Functions• Trigonometric and Inverse trigonometric
Functions• Exponential Functions• Logarithmic Functions
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Function’s Properties 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
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Symmetry
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Transformations of Functions
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Combinations of Functions
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Composition of Functions
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Power Functions
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Exponential Functions
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Inverse Functions
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Logarithmic Functions
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The logarithm with base is called the natural logarithm and has a special notation:
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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.
Inverse Trigonometric Functions
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