1.9. Composite function.

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En provenance du cours de National Research University Higher School of Economics

Mathematics for economists

9 notes

National Research University Higher School of Economics

Mathematics for economists

9 notes

This course is an important part of the undergraduate stage in education for future economists. It's also useful for graduate students who would like to gain knowledge and skills in an important part of math. It gives students skills for implementation of the mathematical knowledge and expertise to the problems of economics. Its prerequisites are both the knowledge of the single variable calculus and the foundations of linear algebra including operations on matrices and the general theory of systems of simultaneous equations. Some knowledge of vector spaces would be beneficial for a student. The course covers several variable calculus, both constrained and unconstrained optimization. The course is aimed at teaching students to master comparative statics problems, optimization problems using the acquired mathematical tools. Home assignments will be provided on a weekly basis. The objective of the course is to acquire the students’ knowledge in the field of mathematics and to make them ready to analyze simulated as well as real economic situations. Students learn how to use and apply mathematics by working with concrete examples and exercises. Moreover this course is aimed at showing what constitutes a solid proof. The ability to present proofs can be trained and improved and in that respect the course is helpful. It will be shown that math is not reduced just to “cookbook recipes”. On the contrary the deep knowledge of math concepts helps to understand real life situations.

À partir de la leçon

The basics of the set theory. Functions in Rn

Week 1 of the Course is devoted to the main concepts of the set theory, operation on sets and functions in Rn. Of special attention will be level curves. Also in this week introduced definitions of sequences, bounded and compact sets, domain and limit of the function. Also from this week students will grasp the concept of continuous function.

1.1. Definitions and examples of sets14:35

1.2. Operations on sets9:31

1.3. Open balls in Rn9:10

1.4. Sequences in Rn. Closed sets.10:33

1.5. Bounded and compact sets10:48

1.6. Functions and level curves in Rn16:47

1.7. Domain and limit of a function. Continuous functions.13:51

1.8. Continuity of a function. Weierstrass theorem.13:04

1.9. Composite function.12:07

1.10. Continuity of a composite function3:30

Rencontrer les enseignants

Kirill Bukin
Associate Professor, Candidate of sciences (phys.-math.)
Faculty of Economic Sciences, Department of Theoretical Economics

Now, we'll explore what will happen if we have a pair of continuous functions.

So, we have a function which is f of x and another is g of x.

Let's suppose that zero belongs to D,

and this is the common domain,

for them both common domain

for f and g,

and we know they are

continuous at x zero.

Can we claim that,

if we form the sum or difference

or product or even a quotient.

Although, in the last case,

we need to exclude while given a condition

will be given g at x zero,

takes a non-zero value.

What can we say? Plus minus a product or quotient.

So, all in all we have four cases here.

Whether they're resulting functions will be continuous at the given point.

Yes that's true and that's fairly easy to prove.

For example, as an example,

we can a choose a product for example,

and how to prove that the limit of the product

as x tends to x zero equals.

In order to prove that, let's recall that,

the definition of a limit of a function of many variables is based

on the limit of the corresponding sequence.

As a sequence of points tends

to x zero and we choose any such sequence.

We consider whether the product of two sequences

has the limit but from

the knowledge of a single variable calculus we now given two convergent sequences.

The first converges to one limit,

the second to another limit,

the product or the terms will converge to the product of

the limits and that concludes the proof of this statement.

As for the quotient it takes

and some more efforts should be placed on showing that,

if the denominator doesn't equal zero at the x zero point.

Then there is a neighborhood or this point where g doesn't equal zero.

So, not only the function itself doesn't turn zero

at a point but also it doesn't turn in some neighborhood,

and that requires some extra statements some work should be done on that.

Now, we are approaching a notion of a composite function.

Composite function, what it means?

I'll start with some examples.

For example, a function f,

depending on xy, sine x times y.

We see that actually there are two functions are involved here,

in the definition or such function because we have a function let us be

g of xy which is x times y.

We have another function which is a function f,

is sin t. Now,

if we say t equals this function g,

which is xy, and we substitute t into here,

so we have a couple of functions,

this one is an inner function sine,

t is an outer function.

When we get the function which was defined by the formula,

it looks as if that we need to think about the domains of two functions.

So, the inner function,

now we'll generalize so that was just an example where let's generalize.

So, we have a function let it

be t equals g of xy,

now f of xy

will be a function h of t. Okay?

So, if we substitute this inner function into h of t,

we get this composite function f. In order to do that,

we need to be sure that the image that

the range over g range of g. Remember,

we get the range of g when we choose any point from the domain of the inner function,

we substitute the values of the point the point into the argument.

As a result we get t value and all such t-values,

they form the range of g. So,

we need to be sure that the domain or the outer function which is a function of

one variable includes the range of g. So,

here I can draw a diagram,

so we have D. D is domain of g. Now,

here we have range of g,

these are the t-values,

and all these values belong to the domain of

h of t. Then

we're sure that we can choose any point from D,

substitute into the argument of g and calculate,

firstly t-value then substitute t-values into h and get

the resulting value of the function f. If not,

we maybe in a complicated situation look at this,

let's consider a function which is provided by the formula minus one,

minus x squared minus y squared.

What about the domain of the inner function?

We can substitute for x and y any values.

So, the domain D coincides with whole a coordinate plane r two,

but the range of this inner function,

so this is our g of xy,

this one, g xy.

From it's the part of the real number line,

but g takes the values which are less than or equal to negative one.

Since we are unable to find the square root of negative values,

social functions simply doesn't exist.

So, the domain for this function is the empty set.

Now, we will be interested in effect,

given a composite function based on a couple of

functions in a function is a function of many variables

like g and the outer function is a function of one variable only h of

t. Can we claim that the composite function based on these two functions,

will be a continuous function given the knowledge that both g of xy and h of

t are continuous at the corresponding points? We'll see.

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