Question

give an example of a function of several variables, but without using a mathematical formula. Can you think of a real life example of something which depends on two, three, or more things? Dont use textbook examples. This is more of a concept type of question. Please write in complete sentences

Answer 1

NOTE: We introduce functions that take vectors or points as inputs and output a number.

The world is constantly changing. Sometimes this change is very slow, other times it is shockingly fast. Consider Meteor Crater in northern Arizona.

This area was once grasslands and woodlands inhabited by bison, camels, woolly mammoths, and giant ground sloths. During the Pleistocene epoch, a meteor only 4040 meters in diameter collided with the Earth and this changed very quickly. The collision released around 4×10164×1016 joules of energy, comparable to the energy released by a large nuclear weapon. A fireball extended out 1010 kilometres from the centre of the impact, destroying all life in its wake. It is estimated it took one hundred years for the local plant and animal life to repopulate the area. Fifty thousand years later, the remains of the impact crater are still intact on our ever-changing Earth.

To help us understand events like these, we need to precisely describe what we are observing (in this case, the crater). To do this we use a contour map, often called a topographical map:

In essence, we are looking at the crater from directly above, and each curve in the map above represents a fixed, constant height. Mathematically, a contour map illustrates a function of two variables. We will now define a more general case of a function of n variables. These are often called functions of several variables.

Let D be a subset of R_n. A function F of n variables, also called a function F of several variables, with domain D is a relation that assigns to every ordered n-tuple in D a unique real number in R. We denote this by each of the following types of notation.

F:D:→R

x↦y

(x1,x2,…,xn)↦y

The range of F is the set of all outputs of F. It is a subset of R, not R_n.

Consider an example:

F:R²→R

(x,y)↦x²+y².

Find the domain and range of F

Here, the domain is

—?

and the range is

—?

The relationship from the previous example can be described more succinctly by the equation

F(x,y)=x²+y²

which is the notation that we will use most frequently when describing functions.

Often, we will not specify the domain of a function in order to shorten its description. Unless otherwise specified, we will take the domain of a given function on Rⁿ to be the set of all ordered tuples in Rⁿ for which the given formula is defined. We are familiar with this concept from one-variable calculus, where we would see a function defined by a formula such as f(x)= √x and take its domain to be [0,∞). In our example F(x,y)=x²+y², we take its domain to be R²