ch00_03

# ch00_03 - 0-29 SECTION 0.3 Inverse Functions 29(a f(0.4(b...

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0-29 SECTION 0.3 . . Inverse Functions 29 (a) f (0 . 4) , (b) f (0 . 39) , (c) f (1 . 17) , (d) f (1 . 20) , (e) f (1.8), (f) f (1.81). Repeat (c)–(d) if the graphing window is zoomed in so that x = 1 . 00 , 1 . 01 ,..., 1 . 20 and y = 1 . 30 , 1 . 31 1 . 50 . Repeat (e)–(f) if the graphing window is zoomed in so that x = 1 . 800 , 1 . 801 1 . 820 and y = 3 . 200 , 3 . 205 3 . 300 . 2. Graph y = x 2 1 , y = x 2 + x 1 , y = x 2 + 2 x 1 , y = x 2 x 1 , y = x 2 2 x 1 and other functions of the form y = x 2 + cx 1 . Describe the effect(s) a change in c has on the graph. 3. Figures 0.31 and 0.32 provide a catalog of the possible types of graphs of cubic polynomials. In this exercise, you will compile a catalog of graphs of fourth-order polyno- mials (i.e., y = ax 4 + bx 3 + 2 + dx + e ) . Start by using your calculator or computer to sketch graphs with different values of a , b , c , d and e . Try y = x 4 , y = 2 x 4 , y =− 2 x 4 , y = x 4 + x 3 , y = x 4 + 2 x 3 , y = x 4 2 x 3 , y = x 4 + x 2 , y = x 4 x 2 , y = x 4 2 x 2 , y = x 4 + x , y = x 4 x and so on. Tryto determine what effect each constant has. f g x Domain { f } y Range { f } FIGURE 0.41 g ( x ) = f 1 ( x ) y x 8 4 2 ± 2 ± 4 6 2 1 ± 2 y ² x 3 FIGURE 0.42 Finding the x -value corresponding to y = 8 0.3 INVERSE FUNCTIONS The notion of an inverse relationship is basic to many areas of science. The number of common inverse problems is immense. As only one example, take the case of the electro- cardiogram (EKG). In an EKG, technicians connect a series of electrodes to a patient’s chest and use measurements of electrical activity on the surface of the body to infer something about the electrical activity on the surface of the heart. This is referred to as an inverse prob- lem, since physicians are attempting to determine what inputs (i.e., the electrical activity on the surface of the heart) cause an observed output (the measured electrical activity on the surface of the chest). The mathematical notion of inverse is much the same as that just described. Given an output (in this case, a value in the range of a given function), we wish to ﬁnd the input (the value in the domain) that produced that output. That is, given a y Range { f } , ﬁnd the x Domain { f } for which y = f ( x ) . (See the illustration of the inverse function g shown in Figure 0.41.) For instance, suppose that f ( x ) = x 3 and y = 8 . Can you ﬁnd an x such that x 3 = 8? That is, can you ﬁnd the x -value corresponding to y = 8? (See Figure 0.42.) Of course, the solution of this particular equation is x = 3 8 = 2 . In general, if x 3 = y , then x = 3 y .In light of this, we say that the cube root function is the inverse of f ( x ) = x 3 . EXAMPLE 3.1 Two Functions That Reverse the Action of Each Other If f ( x ) = x 3 and g ( x ) = x 1 / 3 , show that f ( g ( x )) = x and g ( f ( x )) = x , for all x . Solution For all real numbers x , we have f ( g ( x )) = f ( x 1 / 3 ) = ( x 1 / 3 ) 3 = x and g ( f ( x )) = g ( x 3 ) = ( x 3 ) 1 / 3 = x . ± Notice in example 3.1 that the action of f undoes the action of g and vice versa. We take this as the deﬁnition of an inverse function. (Again, think of Figure 0.41.)

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ch00_03 - 0-29 SECTION 0.3 Inverse Functions 29(a f(0.4(b...

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