Functions Flashcards
Function
A function f is a rule that assigns to each element x in a set D exactly one element, called f(x), in a set E.
What makes a function well-defined?
Well-defined functions must have a single output value for any given input value
Domain
The set D is called the domain of f (the ‘input values’).
What is the domain if the domain is not explicitly stated?
If the domain is not explicitly stated then the domain is the set of all numbers for which the formula makes sense and defines a real number
Range
The range of f is the set of all possible values of f(x) as x varies throughout the domain (the ‘output values’)
Graphs of functions
If f is a function with domain D then its graph is the set of ordered pairs
{(x,f(x))|x ∈ D}
so the graph of f consists of all points (x,y) in the coordinate plane such that y = f(x) and x is the domain of f.
Piecewise function
If a function is defined by different formulas in different parts of its domain then it is called a piecewise function
Even function
If a function f satisfies f(−x) = f(x) for every number x in its domain, then f is called an even function.
Odd function
If a function f satisfies f(−x) = −f(x) for every number x in its domain, then f is called an odd function.
Increasing function
A function f is increasing on an interval I if f (x1) < f (x2) whenever x1 < x2 in I.
Decreasing function
A function f is decreasing on an interval I if f(x1) > f(x2) whenever x1 < x2 in I.
Polynomial
A function P is a polynomial if
P(x) = anx^n+ a(n−1)x^(n−1) + … + a2x^2 + a1x + a0
where n is a nonnegative integer and the numbers a0, a1, a2,…,an are constants
Degree of a polynomial
If the leading coefficient an does not equal 0 then the degree of the polynomial is n.
Power function
A function of the form f(x) = x^a, where a is constant is called a power function.
Root function
A power function (f= x^a) where a = 1/n, where n is a positive integer is a root function
