In general, the graph of a constant function f : x + c is straight line which is parallel to the xaxis at a distance of 1 c ( units from it.
Real N u m b e r a n d Functior~h
2) The identity function: Another simple but important example of a funct~onis a function which sends every element of the domain to itself.
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Let X be any non-empty set, and let f be the function on X defined by setting f(x) = x V x E X.
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This function is known as the identity function on X and is denoted by ix.
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The graph of i, the identity function on R, is shown in Fig. 4. It is the line y = x.
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3) Absolute value Function: Another interesting function is the absolute value funct~on(or modulus function) w h ~ c hcan be defined by using the concept of the absolute value of a real number as: f(x) = I X I =
Fig. 4
x, i f x 2 O - x, if x c 0
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The graph of this function is shown in Fig. 5. It consists of two rays, both starting at the orig~n andmaking angles 7d4 and 3x/4, respectively, with the positive direct~onof the x-axls.
E E 6) Given below are the graphs of four functions depending on the notion of absolute value. r
The functions are x +-I x 1, x + ( x j + 1, x -+ / x + I 1, x + ( x- 1 1, thoughnot necessarily in this order. (The domain in each case is R). Can you identify them?
Fig. 5
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4) The Exponential Vunction: If a is a positive real number other than I , we call define a function f as:
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f R 4 R f (x) 2
(a>O,a+l)
This function is known as the exponential function. A special case of this flmction, where a c , is often found useful. Fig. 6 shows the graph of the function f : R i R such tilai f(x) - e x . This function is also called the natural expo~lrntialfunction. Its range is the set R'of positive real numbers.
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5) The Natural Logarithmic Function: This futictia~:is defined on thc set R ofpositive real numbers, with f:R'+4 R such that f(sj iil (xj. The range of this function is R . Its grnp!, i: shown in Fig. 7.
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Fig. 6
YA = 117s
6) The Greatest Integer Function: Take a real number x. Either it is an integer, say n (so that x = 11) or it is not an intzger. If if is not nil integer, we can find I by the Archimedea~iproperty of real numbers, i ~ nin~cgern. such that n < x <. n t I . Therefore. for each real number x \ve t;~!l find an integer n: such that n I x c: n + 1. Further. for a given real number x; we can f ~ n donly one such integer n. We say that n is the greatest integer not exceeding x, and denotc it by [x]. For example. [?I 3 and 13.51 - 3. (-3.51 = 4.Let 11s consider the function defined on R by Yetrmg f ( x ) -- [XI.
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