Techniques of Algebra Work With Functions
Recall the following from algebra I and see how these laws create new functions.
 (x^{a})(x^{b}) = x^{a+b}  to multiply, add exponents

 x^{a} ÷ x^{b} = x^{ab}  to divide, subtract exponents
Multiplication produces these new functions.
 (constant)(constant) = (constant)
Ex. (5)(2) = 10
 (constant)(line) = (line)
Ex. (5)(2x+3) = 10x+15
 (line)(line) = (quadratic)
Ex. (5x)(x+3) = 5x²+15x
 (line)(quadratic) = (cubic)
Ex. (5x)(x²+3x+1) = 5x^{3}+15x^{2}+5x
 (3rd degree function)(5th degree function) =
(8th degree equation)
Ex. (5x^{3})(2x^{5}) = 10x^{8}
Division produces these new functions.
 (constant)/(constant) = (constant)
Ex. (5)/(2) = 2.5
 (constant)/(line) = (reciprocal function) , also a rational function
Ex. (5)/(2x+3) = (5)/(2x+3)
 (line)/(line) = (reciprocal function), also a rational function
Ex. (5x)/(x+3) = 15/(x+3)  5
 (line)(quadratic) = (rational function)
Ex. (5x)/(x²+3x+1)
 (quadratic)/(line) = (rational function)
Ex. (x²+3x+1)/(5x)
 (3rd degree function)/(5th degree function) =
(rational function)
Ex. (5x^{3})/(2x^{5}) = 5/2x^{2}
When using multiplication to compose a function,
 a zero of a factor is a zero of the composed product function.
 a restriction in a factor is a restriction of the composed product function.
When using division to compose a function,
 almost always, a zero of the numerator is a zero of the composed rational function.
 almost always, a zero of the numerator creates a vertical asymptote of the rational function.
 if the same zero exists in a numerator factor and a denominator factor, a discontinuity
(a hole in the function) is created.
