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These curves are exactly reversed left-to-right from those in Fig 14-1 The above reciprocal functions can be rewritten, respectively, as y = e x and y = 10 x When we negate x before taking the power of the exponential base, we horizontally mirror all of the function values The y axis acts as a point reflector The overall domains and ranges of these reciprocal functions are the same as the domains and ranges of the original functions
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Exponential: example 3 Now that we have two pairs of exponential functions, let s create two new functions by adding them, and see what their graphs look like The solid black curve in Fig 14-3A is a graph of
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y = ex + 1/ex = ex + e x The solid black curve in Fig 14-3B is a graph of y = 10x + 1/10x = 10x + 10 x The domains of these sum functions both encompass all the real numbers The ranges are limited to the reals greater than or equal to 2
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Figure 14-3 Graphs of the natural
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exponential plus its reciprocal (solid black curve at A) and the common exponential plus its reciprocal (solid black curve at B) The dashed gray curves are the graphs of the original functions
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Exponential and Logarithmic Curves
Here s a heads up ! In many of the graphs to come, you ll see two dashed gray curves representing functions to be combined in various ways, as is the case in Fig 14-3 But the constituent functions won t be labeled as they are in Fig 14-3 The absence of labels will keep the graphs from getting too cluttered, so you ll be able to see clearly how they relate Also, the lack of labels will force you to think! Based on your knowledge of the way the functions behave, you should be able to tell which graph is which without having them labeled Exponential: example 4 Figure 14-4 shows what happens when we subtract the reciprocal of the natural exponential function from the original function and then graph the result The solid black curve is the graph of
y = ex 1/ex = ex e x
Figure 14-4 Graph of the natural
exponential minus its reciprocal (solid black curve) The dashed gray curves are the graphs of the original functions
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Figure 14-5 Graph of the
common exponential minus its reciprocal (solid black curve) The dashed gray curves are the graphs of the original functions
In Fig 14-5, we do the same thing with the common exponential function and its reciprocal Here, the solid black curve represents y = 10x 1/10x = 10x 10 x In both of these figures, the dashed gray curves represent the original functions The domains and ranges of both difference functions include all real numbers
Are you confused
Do you wonder how we arrived at the graphs in Figs 14-3 through 14-5 We can plot sum and difference functions in two ways We can graph the original functions separately, and then add or subtract their values graphically (that is, geometrically) by moving vertically upward or downward at various points within the spans of the domains shown Alternatively, we can, with the help of a calculator, plot several points for each sum or difference function after calculating the outputs for several different input values Once we have enough points for the sum or difference function, we can draw an approximation of the graph for that function directly
Exponential and Logarithmic Curves
Here s a challenge!
Plot a graph of the function we get when we raise e to the power of 1/x In rectangular xy coordinates, the curve is represented by the equation y = e(1/x) What is the domain of this function What is its range
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