By J. B. Phear
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Extra info for Elementary Hydrostatics - With Numerous Examples
Show that the line determined by (a, b) and (5, 0) is perpendicular to the line determined by (a, b) and ( - 5, 0). Assume that b -::t 0. Find the length of the common chord of the circles (b) (x - b)2 + ( y - a)2 R 2 (a) (x - a)2 + ( y - b)2 R 2 Write an equation for a line that goes through the point (A , 0) and is tangent to the circle x2 + y2 1 . Assume A > 1 . = = = 1. 7 FUNCTIONS + l Y J = l . ) Find an equation for the unique circle that contains the points (0, - 2) , (6, - 12), and ( - 2, - 4).
Real number and the ordered pair (x, y) is on the line. We generalize this idea in Let X and Y be sets of real numbers. A function number x. range X is called the is called the domain range/ image f is a rule that assigns to each number x in X a single number/(x) in Y. The number/(x) is called the of the off, written domf The set of images of elements of X off This is written orf(X). REMARK. " For example, if f is the function f = 0, the number 0, so that the domain of is the set of all real numbers except zero.
On the other hand, if - 1 < u < l , then u2 - I < 0. But u2 - 1 > - l so that l/(u2 - 1) < - L Thus R - ( - 1, O]. 4). -I g= x. f(x) x y= y2 = x x= EXAM PLE 4. Consider the equation y 2 = The ruk = y where does not determine y as a function of since for every > 0 there are two values yx. For example, if of y such that y 2 namely, y = yx and 4, However, if we specify one of these then 2 and y 2 both satisfy y2 values, say yx, then we have a function. Here dam R+ and range R+. We could obtain a second function by choosing the negative square root.