By Garrett Birkhoff

This vintage, written by means of younger teachers who grew to become giants of their box, has formed the knowledge of recent algebra for generations of mathematicians and continues to be a useful reference and textual content for self learn and school classes.

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In the case of 1/x, the formula does not make sense when x = 0, because division by 0 is not deﬁned. Thus the domain for this function is all real numbers except 0. In this case the range is also all real numbers except 0, but even for relatively simple formulas, the range may be difﬁcult to determine. 1 Functions Given by Formulas √ As another example, consider f (x) = x. This formula does not make sense as a real number if x is a negative number, so the domain is all real numbers greater than or equal to 0.

6 20. Reynolds number: The Reynolds number is very important in such ﬁelds as ﬂuid ﬂow and aerodynamics. In the case of a ﬂuid ﬂowing through a pipe, the Reynolds number R is given by R= 19. Thermal conductivity: The heat ﬂow Q due to conduction across a rectangular sheet of insulating material can be calculated using Q= k (t1 − t2 ) .

1 Functions Given by Formulas E-7. Onto (surjective) functions: A function f : D → R is said to be onto, or surjective, if each element of R is a function value of some element of D. Thus, for example, if D and R are both the collection of all real numbers, then f (x) = x + 1 is onto since given a number y in the range, we have f (y − 1) = (y − 1) + 1 = y. Thus every element of the range is a function value. On the other hand, f (x) = x 2 is not onto since −1 is not the function value of any real number.