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# 4 draw the graphs of the functions y p a 2 x 2 y b p

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4. Draw the graphs of the functions y = p a 2 - x 2 , y = b p 1 - ( x 2 /a 2 ) . 27. D. Implicit Algebraical Functions. It is easy to verify that if y = 1 + x - 3 1 - x 1 + x + 3 1 - x ,

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[II : 27] FUNCTIONS OF REAL VARIABLES 58 then 1 + y 1 - y 6 = (1 + x ) 3 (1 - x ) 2 ; or if y = x + q x + x, then y 4 - (4 y 2 + 4 y + 1) x = 0 . Each of these equations may be expressed in the form y m + R 1 y m - 1 + · · · + R m = 0 , (1) where R 1 , R 2 , . . . , R m are rational functions of x : and the reader will easily verify that, if y is any one of the functions considered in the last set of examples, y satisfies an equation of this form. It is naturally suggested that the same is true of any explicit algebraic function. And this is in fact true, and indeed not difficult to prove, though we shall not delay to write out a formal proof here. An example should make clear to the reader the lines on which such a proof would proceed. Let y = x + x + p x + x + 3 1 + x x - x + p x + x - 3 1 + x . Then we have the equations y = x + u + v + w x - u + v - w , u 2 = x, v 2 = x + u, w 3 = 1 + x, and we have only to eliminate u , v , w between these equations in order to obtain an equation of the form desired. We are therefore led to give the following definition: a function y = f ( x ) will be said to be an algebraical function of x if it is the root of an equation such as (1) , i.e. the root of an equation of the m th degree in y , whose coefficients are rational functions of x . There is plainly no loss of generality in supposing the first coefficient to be unity.
[II : 27] FUNCTIONS OF REAL VARIABLES 59 This class of functions includes all the explicit algebraical functions considered in § 26 . But it also includes other functions which cannot be expressed as explicit algebraical functions. For it is known that in general such an equation as (1) cannot be solved explicitly for y in terms of x , when m is greater than 4, though such a solution is always possible if m = 1, 2, 3, or 4 and in special cases for higher values of m . The definition of an algebraical function should be compared with that of an algebraical number given in the last chapter ( Misc. Exs. 32). Examples XIV. 1. If m = 1, y is a rational function. 2. If m = 2, the equation is y 2 + R 1 y + R 2 = 0, so that y = 1 2 {- R 1 ± q R 2 1 - 4 R 2 } . This function is defined for all values of x for which R 2 1 = 4 R 2 . It has two values if R 2 1 > 4 R 2 and one if R 2 1 = 4 R 2 . If m = 3 or 4, we can use the methods explained in treatises on Algebra for the solution of cubic and biquadratic equations. But as a rule the process is complicated and the results inconvenient in form, and we can generally study the properties of the function better by means of the original equation. 3. Consider the functions defined by the equations y 2 - 2 y - x 2 = 0 , y 2 - 2 y + x 2 = 0 , y 4 - 2 y 2 + x 2 = 0 , in each case obtaining y as an explicit function of x , and stating for what values of x it is defined.

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• Fall '14

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