Multivariate functions: Basic notions
Functions of two variables
Intuitive definition of a function of two variables
So far we have limited ourselves to functions of a single variable. These can be viewed as little machines producing a new number from a given number according to a given formula. But you can also think of little machines and formulas that produce a new number from two given numbers. In that case, we speak of a function of two variables.
Recall from the theory Functions that a function from a set #X# to a set #Y# assigns a unique element of #Y# to each element of #X#. "Functions" in this course are almost always understood to be "real functions". This means #Y=\mathbb R#, both for real functions of a single variable and for functions of two variables.
For real functions of a single variable, #X# is a subset of #\mathbb R#. In the case of a function of two variables, #X# is a subset of #{\mathbb R}^2#, the set of all pairs of real numbers, and #Y=\mathbb R#. Thus, an element of the domain belongs to #{\mathbb R}^2# and is usually denoted by its coordinates #\rv{x,y}#.
You may wonder whether there is a notion corresponding to a little machine that produces two numbers (instead of a single number) from two given numbers. In fact, there is: a function from #{\mathbb R}^2# to #{\mathbb R}^2#.
Three simple functions of two variables
- The area \(O\) of a triangle with base \(b\) and height \(h\): \[O(b,h)=\tfrac{1}{2}\cdot b\cdot h\tiny.\]
- The distance \(s\) covered by an object in uniform motion with velocity \(v\) and time \(t\): \[s(v,t) = v\cdot t\tiny.\]
- The milk consumption #x# as a function of the price #p# of milk and the average income #m# per family: \[x(m,p)=3\cdot \frac{m^{2.07}}{p^{1.4}}\tiny.\]
These examples are written in the form of a function rule, that is a description of the actual machine producing the new number from the given pair #\rv{b,h}# in the first example and #\rv{v,t}# in the second example.
In the second example, the function rule is the expression \(v\cdot t\). The dependent variable #s# is explicitly written down, isolated from the independent variables #v# and #t#.
The terminology of functions which we already know is also used for functions of two variables.
Functions of two variables
A relation between three variables \(x\) , \(y\), and \(z\), where \(x\) and \(y\) occur as independent variables, is a function \(z=z(x,y)\) if, for any admissible values \(x\) and \(y\), there is exactly one value for \(z\) corresponding to it. This value #z# is called the value of the function at the point #\rv{x,y}#.
The set of all pairs#\rv{x,y}# of admissible values \(x\) and \(y\) is called the domain of the function. If #D# is the domain, we speak of a function on #D#.
The value of \(z\) at a point #\rv{x,y}# of the domain is called the function value at #\rv{x,y}#.
The set of all values that the function can assume is called the range of the function.
The graph of a relation between three variables is the set of all points #\rv{x,y,z}# satisfying the relation. In particular, the graph of a function #f# of two variables is the set of all points #\rv{x,y,f(x,y)}# for #\rv{x,y}# ranging over the points of the domain of #f#.
The definition of graph is a straightforward generalization of the notion of graph given in the case of a single variable.
Substituting #a=2# and # b=1# in the expression # (2{a}^2+{b})^2-9{a}\cdot{b}-9{b}# gives \[ (2\cdot (2)^2+(1))^2-9\cdot (2)\cdot (1)-9\cdot (1)\tiny.\] Simplification of this expression gives the answer #54#.
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