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Asymptote along y axis
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Range includes all positive real numbers except 1
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Asymptote at y = 1
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Domain includes all nonzero real numbers
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Figure B-17 Illustration for the solution to Problem 5 in Chap 14
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Let s take the natural exponential of each side of this equation That gives us e(ln 0) = e x We know that the natural log function and the natural exponential function are inverses of each other They undo each other s work, as long as we stay within the domain of the natural log function We ve assumed that ln 0 is a real number, and therefore that it s in the domain of the natural log function Based on that assumption, we can rewrite the above equation as 0 = ex In the solution to Problem 1, we proved that no real number x can satisfy this equation That result contradicts our original assumption here By reductio ad absurdum, we are forced to conclude that the natural log of 0 is not a real number 7 The dashed gray curves in Fig B-18 are the graphs of y = ln x and y = log10 x We want to graph the sum function y = ln x + log10 x When we input several values, use a calculator to obtain the outputs, plot the points, and then connect the points by curve fitting, we get the solid black curve The domain of this sum function is the set of positive reals, and the range is the set of all reals Figure B-18
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Illustration for the solution to Problem 7 in Chap 14
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544 Worked-Out Solutions to Exercises: 11-19
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Figure B-19
Illustration for the solution to Problem 8 in Chap 14
8 The dashed gray curves in Fig B-19 are the graphs of y = ln x and y = log10 x The portions of these curves below the x axis (that is, where y < 0) are cut off here, because we haven t included any of the negative y axis But the function values are still there, of course! We want to graph the product function y = (ln x)(log10 x) When we input several values for x, use a calculator to obtain the outputs, plot the points, and then connect the points by curve fitting, we get the solid black curve The domain of this product function is the set of all positive reals, and the range is the set of all nonnegative reals 9 Figure B-20 shows the same graphs as Fig B-18 Here, the x axis is logarithmic, spanning the two orders of magnitude from 01 to 10 The dashed gray lines are the graphs of y = ln x and y = log10 x The solid black line is the graph of the sum function y = ln x + log10 x
14
Figure B-20
Illustration for the solution to Problem 9 in Chap 14
03 0 1 3 10 x
10 Figure B-21 shows the same graphs as Fig B-19 In this coordinate system, the x axis is logarithmic, spanning the single order of magnitude from 1 to 10 The y axis is linear, spanning the values from 0 to 25 The dashed gray lines are the graphs of y = ln x and y = log10 x The solid black curve is the graph of the product function y = (ln x)(log10 x) Figure B-21
Illustration for the solution to Problem 10 in Chap 14
x 1 3 10
546 Worked-Out Solutions to Exercises: 11-19
15
1 Figure B-22 shows superimposed graphs of the sine and cosine functions (dashed gray curves) along with a graph of their difference function h (q) = sin q cos q shown as a solid black curve The domain of h includes all real numbers The range of h is the set of all reals in the closed interval [ 21/2,21/2] 2 The solid black curve in Fig B-23 is a graph of the difference between the squares of the sine and the cosine functions, which are shown as superimposed dashed gray curves We have h (q) = sin2 q cos2 q The domain of h includes all of the real numbers The range is the set of all real numbers in the closed interval [ 1,1]
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