By Stanley I Grossman
1,178 pages plus Appendixes of 146 pages
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The graph of this function is obtained by plotting all points of the form (x, y) = (x, x 2). t First, we note that because f(x) = x 2, f( ) = /( - since ( -x)2 = x 2 • Thus it is only necessary to calculate f(x) for x � 0. For every x > 0 there is a value of x < 0 (the number -x) that gives the same value of y. In this situation we say that the function is symmetric about the y-axis. Some values for f(x) are shown in Table This function, which is graphed in Figure 1, is called a parabola. 5)]. O} x x) 1.
F(x) is Definition 3 SYMMETRIC FUNCTION The function given by y metric about the y-axis if f(x) = f( - x). sym- A function that is symmetric about the y-axis is also called an even function. At this point there are essentially three reasons for restricting the domain of a function: (i) You cannot have zero in a denominator. ) of a negative number. (iii) The domain is restricted by the nature of the applied problem under con sideration (see Example 7). 2. EXAMPLE 4 Let f(x) = v'2x - 6. Find the domain of f.
Then dom g = [O, oo) , dom f n dom g = [O, oo) . But ( f • g)(x) = Vx 2Vx = 2x, which is defined for every real number x -or is it? The apparent problem here lies in the definition of a function. The function fg , with domain [O, oo), is defined as the rule ( fg )(x ) = 2x for x � 0. This function is not the same function as the function defined by h (x) = 2x without the restriction that x � 0. These two functions are not the same since they have different domains. They also have different graphs.