Functions and Calculus, Part 3

Question 1: Describe the differences between algebraic functions and transcendental functions.

Answer 1: Algebraic functions are those that exclusively use polynomials and roots. These would include polynomial functions, rational functions, square root functions, and all combinations of these functions, such as polynomials as the radicand. These combinations may be joined by addition, subtraction, multiplication, or division, but may not include variables as an exponent.Transcendental functions are all functions that are non-algebraic. Any function that includes logarithms, trigonometric functions, variables as exponents, or any combination that includes any or all of these is not algebraic in nature, even if the function includes polynomials or roots, and therefore a transcendental function.

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Question 2: Describe equal functions and explain how to find the sum or difference of two functions.

Answer 2: Equal functions are those whose domains are equal, and whose ranges are equal for all corresponding values in the domain. In other words, f(x) and g(x) are equal if every value of f(x) is equal to every corresponding value of g(x).To find the sum of the functions f and g, assuming the ranges are all real numbers, solve each function individually and add the results: ?+?=?+?(?).To find the difference of the functions f and g, assuming the ranges are all real numbers, solve each function individually and subtract the results: ?-?=?-?(?

Question 3: Explain how to find the product, quotient, and composite of two functions.

Answer 3: To find the product of the functions f and g, assuming the ranges are all real numbers, solve each function individually and then multiply the results: ?·?=?·?(?). This is much easier, and less prone to mathematical error, than trying to multiply two polynomials together before applying the value of x.To find the quotient of the functions f and g, assuming the ranges are all real numbers, solve each function individually and then divide the results: ?=? ;?0.The composite of two functions f and g, represented by the symbol ?°?(?) or ?, is found by substituting ?(?) for all instances of x in f(x) and simplifying. It is important to note that ?°?(?) does not always equal ?°?(?). The process is not commutative like addition or multiplication expressions. If ?°?(?) does equal ?°?(?), the two functions are inverses of each other. This is one of the easiest tests to determine if two functions are inverses. If the two functions are graphed, the graphs will be reflections of each other with respect to the line y = x.

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