FUNctions and Their Graphs Chapter One

1.1 FUNctions How can I evaluate FUNctions and find their domains?

Leonhard Euler Portrait by Johann Georg Brucker Born April 15April 15, 1707( ) Basel, Switzerland1707 BaselSwitzerland Died September 18September 18 [O.S. September 7] 1783 St Petersburg, RussiaO.S.1783 St PetersburgRussia Residence Prussia Russia Switzerland NationalitySwiss FieldMathematicsMathematics and physicsphysics Institutions Imperial Russian Academy of Sciences Berlin Academy Alma materUniversity of Basel ReligionCalvinist [1] Picture and info from

Swiss, but spent most of his life in Germany and Russia Preeminent mathematician of the 18 th century and one of the greatest of all time – His works fill over 70 volumes. Described as a kind and generous man – Enjoyed growing vegetables – Enjoyed telling stories to his 13 children Saint in the Lutheran Church (May 24) Famous for work on calculus, and, of course, FUNctions

Lost eyesight with age, but still produced mathematics Francois Arago said he could calculate without effort, “just as men breathe, as eagles sustain themselves in the air” Spent last day playing with his grandchildren and discussing the latest astronomical theories The end came quickly, where mathematician Condorcet said, “he ceased to calculate and live.”

Introduction to FUNctions “To those who ask what the infinitely small quantity in mathematics is, we answer that it is actually zero. Hence there are not so many mysteries hidden in this concept as they are usually believed to be.”

A relationship can be shown in several ways. Ordered Pairs Table xy Graph Map

The domain is (1,2,3,4) The range is (2,4,0,5) The following is an example of a relation, or a set of ordered pairs. Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate.

Please state the domain and range of the following relation: Please review the following example: Domain: {0,2,1,5}Range: {0,1,3,2}

A function is a relationship where every domain element corresponds to one and only one range element. Now I will have less distraction. [upon losing the use of his right eye]

Example What is the domain of S? What is the range of S? Is S a function? yes

Which of the following represents y as a function of x? Yes! No!

FUNction Notation We name functions so that they can be referenced. Popular names for FUNctions are f, g, and h. In addition to the name, we mention the value we are putting into the FUNction. So, we can call a FUNction f. – If we are using x as the variable, we can write f(x). – If we wanted to find the value for a in the equation, we would write f(a).

Remember Also – f is the NAME of the FUNction – it is not another variable.

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The Domain of a FUNction For all polynomials, the domain is Fractions cannot have a value which makes the denominator equal to zero. If the index of a radical is even, then the radicand must be non-negative. “For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear.”

Please find the domain of the following FUNctions.

... for the sake of brevity, we will always represent this number by the letter e.

The Difference Quotient Please find

Try these! -16 Is the following a FUNction? no Please find the domain. “Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena.”