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INVERSE FUNCTIONS

7. INVERSE FUNCTIONS. INVERSE FUNCTIONS. The common theme that links the functions of this chapter is: They occur as pairs of inverse functions. INVERSE FUNCTIONS. In particular, two among the most important functions that occur in mathematics and its applications are:

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INVERSE FUNCTIONS

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  1. 7 INVERSE FUNCTIONS

  2. INVERSE FUNCTIONS • The common theme that linksthe functions of this chapter is: • They occur as pairs of inverse functions.

  3. INVERSE FUNCTIONS • In particular, two among the most important functions that occur in mathematics and its applications are: • The exponential function f(x) = ax. • The logarithmic function g(x) = logax, the inverse of the exponential function.

  4. INVERSE FUNCTIONS • In this chapter, we: • Investigate their properties. • Compute their derivatives. • Use them to describe exponential growth and decay in biology, physics, chemistry,and other sciences.

  5. INVERSE FUNCTIONS • We also study the inverses of trigonometric and hyperbolic functions. • Finally, we look at a method (l’Hospital’s Rule) for computing difficult limits and applyit to sketching curves.

  6. INVERSE FUNCTIONS • There are two possible ways of defining the exponential and logarithmic functions and developing their properties and derivatives. • You need only read one of these two approaches—whichever your instructor recommends.

  7. INVERSE FUNCTIONS • One is to start with the exponential function (defined as in algebra or precalculus courses) and then define the logarithm as its inverse. • This approach is taken in Sections 7.2, 7.3, and 7.4. • This is probably the most intuitive method.

  8. INVERSE FUNCTIONS • The other way is to start by definingthe logarithm as an integral and then define the exponential function as its inverse. • This approach is followed in Sections 7.2*, 7.3*, and 7.4*. • Although it is less intuitive, many instructors prefer it because it is more rigorous and the properties follow more easily.

  9. INVERSE FUNCTIONS 7.1 Inverse Functions • In this section, we will learn about: • Inverse functions and their calculus.

  10. INVERSE FUNCTIONS • The table gives data from an experiment • in which a bacteria culture started with • 100 bacteria in a limited nutrient medium. • The size of the bacteria population was recorded at hourly intervals. • The number of bacteria N is a function of the time t: N = f(t).

  11. INVERSE FUNCTIONS • However, suppose that the biologist changes • her point of view and becomes interested in • the time required for the population to reach • various levels. • In other words, she is thinking of t as a function of N.

  12. INVERSE FUNCTIONS • This function is called the inverse • function of f. • It is denoted by f-1 and read “f inverse.”

  13. INVERSE FUNCTIONS • Thus, t = f-1(N) is the time required for • the population level to reach N.

  14. INVERSE FUNCTIONS • The values of f-1can be found by reading • the first table from right to left or by consulting • the second table. • For instance, f-1(550) = 6, because f(6) = 550.

  15. INVERSE FUNCTIONS • Not all functions possess • inverses. • Let’s compare the functions f and g whose arrow diagrams are shown.

  16. INVERSE FUNCTIONS • Note that f never takes on the same • value twice. • Any two inputs in A have different outputs.

  17. INVERSE FUNCTIONS • However, g does take on the same • value twice. • Both 2 and 3 have the same output, 4.

  18. INVERSE FUNCTIONS • In symbols, g(2) = g(3) • but f(x1) ≠ f(x2) whenever x1 ≠ x2

  19. INVERSE FUNCTIONS • Functions that share this property • with f are called one-to-one functions.

  20. ONE-TO-ONE FUNCTIONS Definition 1 • A function f is called a one-to-one • function if it never takes on the same • value twice. • That is, • f(x1) ≠ f(x2) whenever x1≠ x2

  21. ONE-TO-ONE FUNCTIONS • If a horizontal line intersects the graph of f • in more than one point, then we see from • the figure that there are numbers x1and x2 • such that f(x1) = f(x2). • This means f is not one-to-one.

  22. ONE-TO-ONE FUNCTIONS • So, we have the following • geometric method for determining • whether a function is one-to-one.

  23. HORIZONTAL LINE TEST • A function is one-to-one if and only if • no horizontal line intersects its graph • more than once.

  24. ONE-TO-ONE FUNCTIONS Example 1 • Is the function f(x) = x3 one-to-one?

  25. ONE-TO-ONE FUNCTIONS E. g. 1—Solution 1 • If x1≠ x2, then x13≠x23. • Two different numbers can’t have the same cube. • So, by Definition 1, f(x) = x3 is one-to-one.

  26. ONE-TO-ONE FUNCTIONS E. g. 1—Solution 2 • From the figure, we see that no horizontal • line intersects the graph of f(x) = x3 more • than once. • So, by the Horizontal Line Test, f is one-to-one.

  27. ONE-TO-ONE FUNCTIONS Example 2 • Is the function g(x) = x2one-to-one?

  28. ONE-TO-ONE FUNCTIONS E. g. 2—Solution 1 • The function is not one-to-one. • This is because, for instance, g(1) = 1 = g(-1)and so 1 and -1 have the same output.

  29. ONE-TO-ONE FUNCTIONS E. g. 2—Solution 2 • From the figure, we see that there are • horizontal lines that intersect the graph • of g more than once. • So, by the Horizontal Line Test, g is not one-to-one.

  30. ONE-TO-ONE FUNCTIONS • One-to-one functions are important because: • They are precisely the functions that possess inverse functions according to the following definition.

  31. ONE-TO-ONE FUNCTIONS Definition 2 • Let f be a one-to-one function with • domain A and range B. • Then, its inverse function f-1 has domain B • and range A and is defined by • for any y in B.

  32. ONE-TO-ONE FUNCTIONS • The definition states that, if f maps x • into y, then f -1 maps y back into x. • If f were not one-to-one, then f-1 would not be uniquely defined.

  33. ONE-TO-ONE FUNCTIONS • The arrow diagram in the figure • indicates that f-1 reverses the effect of f.

  34. ONE-TO-ONE FUNCTIONS • Note that: • domain of f -1 = range of f • range of f -1 = domain of f

  35. ONE-TO-ONE FUNCTIONS • For example, the inverse function • of f(x) = x3 is f-1(x) = x1/3. • This is because, if y = x3, then f -1(y) = f -1(x3) = (x3)1/3 = x

  36. ONE-TO-ONE FUNCTIONS Caution • Do not mistake the -1 in f-1 • for an exponent. • Thus, f-1(x) does not mean . • However, the reciprocal could be written as [f(x)]-1.

  37. ONE-TO-ONE FUNCTIONS Example 3 • If f(1) = 5, f(3) = 7, and f(8) = -10, • find f-1(7), f-1(5), and f-1(-10). • From the definition of f-1, we have:f -1(7) = 3 because f(3) = 7f -1(5) = 1 because f(1) = 5f -1(-10) = 8 because f(8) = -10

  38. ONE-TO-ONE FUNCTIONS Example 3 • This diagram makes it clear how f-1 • reverses the effect of f in this case.

  39. ONE-TO-ONE FUNCTIONS Definition 3 • The letter x is traditionally used as the • independent variable. • So, when we concentrate on f-1 rather than • on f, we usually reverse the roles of x and y • in Definition 2 and write:

  40. CANCELLATION EQUATIONS Definition 4 • By substituting for y in Definition 2 and • substituting for x in Definition 3, we get • the following cancellation equations: • f -1(f(x)) = x for every x in A • f(f -1(x)) = x for every x in B

  41. CANCELLATION EQUATION 1 • The first cancellation equation states that, • if we start with x, apply f, and then apply • f-1, we arrive back at x, where we started. • Thus, f-1 undoes what f does.

  42. CANCELLATION EQUATION 2 • The second equation states that • f undoes what f-1 does.

  43. CANCELLATION EQUATIONS • For example, if f(x) = x3, then f-1(x) = x1/3. • So, the cancellation equations become: • f -1(f(x)) = (x3)1/3 = x • f(f -1(x)) = (x1/3)3 = x • These equations simply states that the cube function and the cube root function cancel each other when applied in succession.

  44. INVERSE FUNCTIONS • Now, let’s see how to compute inverse • functions. • If we have a function y = f(x) and are able to solve this equation for x in terms of y, then, according to Definition 2, we must have x = f-1(y). • If we want to call the independent variable x, we then interchange x and y and arrive at the equation y = f-1(x).

  45. INVERSE FUNCTIONS Method 5 • Now, let’s see how to find the inverse • function of a one-to-one function f. • Write y = f(x). • Solve this equation for x in terms of y (if possible). • To express f-1 as a function of x, interchange x and y. The resulting equation is y = f-1(x).

  46. INVERSE FUNCTIONS Example 4 • Find the inverse function of • f(x) = x3 + 2. • By Definition 5, we first write: y = x3 + 2. • Then, we solve this equation for x : • Finally, we interchange x and y : • So, the inverse function is:

  47. INVERSE FUNCTIONS • The principle of interchanging x and y • to find the inverse function also gives us • the method for obtaining the graph of f-1 • from the graph of f. • As f(a) = b if and only if f-1(b) = a, the point (a, b) is on the graph of f if and only if the point (b, a) is on the graph of f-1.

  48. INVERSE FUNCTIONS • However, we get the point (b, a) from • (a, b) by reflecting about the line y = x.

  49. INVERSE FUNCTIONS • Thus, the graph of f-1 is obtained by • reflecting the graph of f about the line • y = x.

  50. INVERSE FUNCTIONS Example 5 • Sketch the graphs of • and its inverse function using the same • coordinate axes.

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