Rekenen met variabelen *: Variabelen *
The notion of function
Functions
Let #X# and #Y# be sets.
A function from #X# to #Y# assigns to each element of #X# one element from the set #Y#.
The set #X# is called the domain of the function. The set #Y# is called the codomain of the function.
The notation #f:X\to Y# indicates that #f# is a function from #X# to #Y#.
The element of #Y# that the function #f# assigns to an element #a# of #X#, is denoted by #f(a)# and is called the value of (the function) #f# at #a# (it is pronounced as: #f# of #a#) or the function value at #a#. This values is also called the image of #a# under #f#.
If we choose a variable #x# for #a# (that is, a symbol representing an arbitrary element of #X#), then #x# is called the argument of #f(x)#.
For instance, if we say that #f# is the function #x^2#, we mean that the function rule of this function is #f(x)=x^2#. The right hand side of this equality indicates what value is assigned to an arbitrary #x#.
For the greater part of this course, the codomain will be #\mathbb{R}# (the real numbers). Such a function is called real valued. If, in addition, the domain is a part of #\mathbb{R}#, then the function is called real.
The function rule is the expression telling us how to calculate the value #f(x)# for each element #x# of #X#. For instance, #f(2)# is the value of the function at #2#, but not the function rule.
The value #f(x)# is often denoted by #y#. In this regard, #y=x^2# refers to the function with rule #x^2#.
The choice for these specific names of the variables #x# and #y# is a habit from which we can deviate whenever we wish. To avoid any confusion it is important to mention the arguments of functions explicitly. For instance, by not simply writing an expression like #a\,x^2+t#, but by specifying, #a\,x^2+t# as a function of #a# (rather than #x# or #t#).
Consider the function #f:\mathbb{R}\to\mathbb{R}# that assigns to #x# the value #x^2#. The function is described by #f(x)=x^2#. Thus, the function rule is #x^2#. The value of the function at #3# is #3^2#, which equals #9#. At #-3#, the value of the function is also #9#.
Often, the function rule #x^2# is also called a function. In this case we refer to the function defined by the rule. Here the domain is #\mathbb{R}#, but, for the function #\dfrac{1}{x}#, there is a problem with the function rule at #0#: the quotient #\dfrac{1}{0}# has no meaning. We say that #\dfrac{1}{x}# is not defined at #0# and interpret #\dfrac{1}{x}# as the function #g:\mathbb{R}\setminus\{0\}\to \mathbb{R}# given by the function rule #g(x) = \dfrac{1}{x}#.
In the function rule, the letter #x# plays a role that can be replaced by any other unused letter (or even longer name). For instance, #a\,x^2+t# can be seen as a function #f# of #a#, of #x#, or of #t#. The choice of, say, #a# as the argument can be indicated by the function rule: #f(a) = a\,x^2+t#. The other variables (#x# and #t# in this example), are called parameters, in order to differentiate from the argument #a# of the function #f#.
Indeed, #f(0) = \frac{0}{{0}-{2}} = 0#.
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