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
Reciprocal function
A power function (f =x^a) where a = −1 (f(x) = 1/x) is a reciprocal function
Range of sin and cos functions
Range is the closed interval [−1, 1].
secx
secx= 1/cosx
cscx
cscx= 1/sinx
cotx
cotx= 1/tanx
Exponential functions
Exponential functions take the form f(x) = b^x where the base, b, is a positive constant.
Logarithmic functions
Logarithmic functions take the form g(x) = logb x where the base, b, is a positive constant.
The natural logarithm
The natural logarithm, ln x = loge x , is the special case where the base is e = 2.71828…, the unique number whose natural logarithm is equal to one.
y = f(x) + c translation
the graph of y = f(x) shifts c units upward
y = f(x) − c translation
the graph of y = f(x) shifts c units downward
y = f(x − c) translation
the graph of y = f(x) shifts c units to the right
y = f(x + c) translation
the graph of y = f(x) shifts c units to the left
How do translations of functions arise?
Translations of functions arise by adding or subtracting a constant, c, to the function or its argument
How does stretching of a function arise?
Stretching of functions arises by multiplying or dividing the function or its argument by a constant, c.
y = cf(x) transformation
the graph of y = f(x) stretches vertically by a factor of c
y = f(x)/c transformation
the graph of y = f(x) shrinks vertically by a factor of c
y = f(cx) transformation
the graph of y = f(x) shrinks horizontally by a factor of c
y = f(x/c) transformation
the graph of y = f(x) stretches horizontally by a factor of c
How do reflections of functions arise?
Reflections of functions arises by multiplying the function or its argument by −1.
y = −f(x) transformation
the graph of y = f(x) reflects about the x-axis
y = f(−x) transformation
the graph of y = f(x) reflects about the y-axis
Composite functions
Given two functions, f and g, the composite function, f ◦ g (also called the composition of f and g) is defined by
(f ◦ g)(x) = f(g(x)).
