MTH 10 Fall 007 Essex County College Division of Mathematics Handout Version 6 1 October, 007 1 Inverse Functions This section is a simple review of inverses as presented in MTH-119. Definition: A function f is called one-to-one if it never takes on the same value twice; that is f x 1 f x whenever x 1 x. We can often visually verified that a function is one-to-one by using the horizontal line test, which states, A function is one-to-one if an only if no horizontal line intersects its graph more than once. For example, the following function is not one-to-one. 1 - -1 0 1 However, this function is one-to-one. Figure 1: f x = x x + 5 1 - -1 0 1 Figure : f x = x + 1 Some however may be mislead by the horizontal line test, because it is hard to detect from the visual alone that f x = x + 1 is one-to-one. It should be clear though that f x = x x + 5 is not one-to-one from the graph alone. To show that f x = x + 1 is one-to-one we proceed as follows. Since for all x 1, x 1, and x 1 x, we have x 1 x, which further implies that x 1 + 1 x + 1, and finally that x 1 + 1 x + 1. 1 This document was prepared by Ron Bannon using L A TEX ε. 1
Definition: Let f be a one-to-one function with domain A and range B. function f 1 has a domain B and range A and is defined by Then its inverse f 1 y = x f x = y. To find the inverse of a one-to-one function, follow these steps. 1. In the equation for f x, replace f x by y.. Interchange the roles of x and y, and solve for y.. Replace y by f 1 x in the new equation. 4. Although not necessary, you may want to verify that f f 1 x = f 1 f x = x. Let s start with a simple example. Given a one-to-one function, f x = x 1, find its domain, range, and inverse. Work: First off we know f is one-to-one with domain x R and range f x R, so its inverse, denoted f 1 x, has a domain x R and range f 1 x R. Using simple algebra method above to find f 1 x. f x = x 1 y = x 1 write y = f x x = y 1 interchange x and y x + 1 = y solve for y x + 1 = f 1 x 4 1-5 -4 - - -1 0 1 4 5-1 - - -4 Figure : f x, and f 1 x in blue. You should observe the symmetry along the line y = x. Here s a more difficult example, however it is still algebraically manageable. Using f x = x + 1, let s find its inverse. First off we know f is one-to-one with domain x 1 and range
f x 0, so its inverse, denoted f 1 x, has a domain x 0 and range f 1 x 1. Using simple algebra to find f 1 x. f x = x + 1 y = x + 1 write y = f x x = y + 1 interchange x and y x = y + 1 solve for y x 1 = y x 1 = y x 1 = f 1 x write y = f 1 x 5 4 1-4 - - -1 0 1 4 5-1 - Figure 4: f x, and f 1 x in blue. You should observe the symmetry along the line y = x. An even more difficult example, given, f x = 1 x 1 + x. find f 1 and its domain and range. A graph is provided as a visual aid.
-1 0 1 1 Figure 5: f x, and f 1 x in blue. Work: We finally have, f x = 1 x 1 + x y = 1 x 1 + x x = 1 y 1 + y x + x y = 1 y y + x y = 1 x y 1 + x = 1 x 1 x y = 1 + x 1 x y = 1 + x f 1 x = and its domain is 1 1] and its range is [0,. 1.1 The Inverse Trigonometric Functions 1 x, 1 + x Important: I will now be writing x instead of θ, so please bear in mind that x now represents the angle! and if I write y = sin x, then y represents the ratio. I know this confuses many students, but it s done in almost all books and it s hard to avoid! The original definitions that I gave you back in the beginning have not changed, but here, x and y are now different! If you ve learned the previous material in MTH-119 you will immediately notice that none of the trigonometric functions are invertible. So we will need to restrict the domain of all six functions to force the conditions of invariability. I ll only do three, starting with the sine function. 4
1.1.1 The Inverse Sine Function 0 Figure 6: The sine function and an invertible region. 1. What is the functional notation of the red portion of this graph?. What is this function s domain?. What is this function s range? 4. The name of sine s inverse is arcsine, some also write inverse sine, or sin 1 x. a What is the arcsine s domain? b What is the arcsine s range? 5
5. Label the following graph. You should indicate the equation of each function there s two and important points along each axis. 0 Figure 7: Please label. 6
1.1. The Inverse Cosine Function 0 Figure 8: Please label. 1. What is the functional notation of the red portion of this graph?. What is this function s domain?. What is this function s range? 4. The name of cosine s inverse is arccosine, some also write inverse cosine, or cos 1 x. a What is the arccosine s domain? b What is the arccosine s range? 7
5. Label the following graph. You should indicate the equation of each function there s two and important points/features along each axis. 0 Figure 9: Please label. 8
1.1. The Inverse Tangent Function 0 - Figure 10: Please label. 1. What is the functional notation of the red portion of this graph?. What is this function s domain?. What is this function s range? 4. The name of tangent s inverse is arctangent, some also write inverse tangent, or tan 1 x. a What is the arctangent s domain? b What is the arctangent s range? 9
5. Label the following graph. You should indicate the equation of each function there s two and important points/features along each axis. 0 - Figure 11: Please label. 10
1. Examples 1. Without using your calculator, evaluate arcsin 1/. What does this number represent?. Without using your calculator, evaluate arcsin 1/. What does this number represent?. Without using your calculator, evaluate arcsin /. 4. Without using your calculator, evaluate arccos /. What does this number represent? 5. Without using your calculator, evaluate arccos 0. What does this number represent? 6. Without using your calculator, evaluate arccos 1. What does this number represent? 7. Without using your calculator, evaluate arctan /. What does this number represent? 8. Without using your calculator, evaluate arctan 1/. What does this number represent? 9. Without using your calculator, evaluate arctan 1. What does this number represent? 10. Using your calculator, evaluate arcsin 0.987. Answer both in radian and degree. 11
11. Using your calculator, evaluate arccos 0.0895. Answer both in radian and degree. 1. Using your calculator, evaluate arctan 4.589. Answer both in radian and degree. 1. Using your calculator, evaluate arccos /4. Answer both in radian and degree. 14. Using your calculator, evaluate arcsin 5/7. Answer both in radian and degree. 15. Using your calculator, evaluate arctan 5.894. Answer both in radian and degree. 16. Evaluate each of the following, and in each case you should clearly indicate if your answer represents a ratio or an angle. a cos 1 sin 6 b sin cos 1 5 c tan sin 1 4 5 d csc cos 1 7 5 e sin 1 sin 6 1
f sin 1 sin 5 6 g cos 1 cos 7 h cos 1 cos i tan 1 tan j cos sin 1 k tan arcsin 1. Answers to Examples 1. Without using your calculator, evaluate arcsin 1/. What does this number represent? 6 or 0, it s an angle. Without using your calculator, evaluate arcsin 1/. What does this number represent? 4 or 45, it s an angle. Without using your calculator, evaluate arcsin /. Not possible! The ratio can not be bigger than one. 4. Without using your calculator, evaluate arccos /. What does this number represent? 5 or 150, it s an angle 6 1
5. Without using your calculator, evaluate arccos 0. What does this number represent? or 90, it s an angle 6. Without using your calculator, evaluate arccos 1. What does this number represent? or 180, it s an angle 7. Without using your calculator, evaluate arctan /. What does this number represent? 6 or 0, it s an angle 8. Without using your calculator, evaluate arctan 1/. What does this number represent? 6 or 0, it s an angle 9. Without using your calculator, evaluate arctan 1. What does this number represent? 4 or 45, it s an angle 10. Using your calculator, evaluate arcsin 0.987. Answer both in radian and degree. 1.409 or 80.751, it s an angle 11. Using your calculator, evaluate arccos 0.0895. Answer both in radian and degree. 1.6604 or 95.148, it s an angle 1. Using your calculator, evaluate arctan 4.589. Answer both in radian and degree. 1.56 or 77.707, it s an angle 1. Using your calculator, evaluate arccos /4. Answer both in radian and degree..4188 or 18.5904, it s an angle 14. Using your calculator, evaluate arcsin 5/7. Answer both in radian and degree. 0.79560 or 45.5847, it s an angle 15. Using your calculator, evaluate arctan 5.894. Answer both in radian and degree. 1.407 or 80.707, it s an angle 16. Evaluate each of the following, and in each case you should clearly indicate if your answer represents a ratio or an angle. a cos 1 sin 6 6 or 0, it s an angle 14
b sin cos 1 5 4, it s a ratio 5 c tan sin 1 4 5 4, it s a ratio d csc cos 1 7 5 5, it s a ratio 4 e sin 1 sin 6 6 or 0, it s an angle f sin 1 sin 5 6 6 or 0, it s an angle g cos 1 cos 7 or 90, it s an angle h cos 1 cos or 90, it s an angle i tan 1 tan j cos or 60, it s an angle sin 1 1, it s a ratio k tan arcsin 1, it s a ratio 15