Engineering Jump Start - Summer 2012 Mathematics Worksheet #4. 1. What is a function? Is it just a fancy name for an algebraic expression, such as

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1 Function Topics Concept questions: 1. What is a function? Is it just a fancy name for an algebraic expression, such as x 2 + sin x + 5? 2. Why is it technically wrong to refer to the function f(x), for example, in statements such as suppose the function f(x) is continuous..? 3. What is a piecewise defined function? Define the concept and give an example. 4. What are domain and range of a function? 5. What do we mean by continuity of a function? 6. What is the meaning of composition of two functions, f composed with g? What condition about the range of g and the domain of f needs to be satisfied so that the composition even exists? 7. What are the domain and range of the composition of f with g? 8. What symbol is commonly used to denote composition of functions? 9. Is f composed with g the same as the product function f times g? If not, explain the difference in definition. 10. What does it mean for a function to be invertible, and what is the meaning of the inverse function? Can you write algebraic equations that make the meaning exact? 11. What is the meaning of the notation f 1? 12. True or false? f 1 (x) means the same as 1 f(x). 13. If the graph of a function is given, how can you tell from the graph whether the function is invertible and how can you then geometrically find the graph of the inverse?

2 14. What is the common algebraic procedure for finding an inverse? 15. What is the inverse of the identity function f(x) = x? 16. Find a linear function (y=mx+b) that s not the identity function which is equal to its own inverse. 17. Find ALL linear functions that are equal to their own inverses. 18. Find a non-linear functions that is equal to its own inverse. 19. Are the trigonometric functions invertible? Consider your answer carefully. 20. How are domain and range of the inverse of a function related to the domain and range of the original function? 21. Is the composition of two invertible functions always invertible? Explain. 22. What do we mean when we say that a function is increasing or decreasing? Try to formulate a precise definition as well as a geometric/visual one. 23. What do we mean when we say that a function is strictly increasing or strictly decreasing? 24. Find an example of a function that is both increasing and decreasing. 25. What connection exists between the concept of a strictly increasing function and invertibility? What about if we drop the strictly part? 26. True or false? If a function is strictly increasing, so is its inverse. Give a geometric and an algebraic justification for your answer. 27. (Only for students who know derivatives) Can a function be strictly increasing and yet have zero derivative at some point(s)? If yes, give an example. If no, explain why not. 28. What can you say about the composition of a strictly increasing function with another strictly increasing function? Can you prove your answer based on the precise definition you formulated in a previous question?

3 29. What can you say about the composition of a strictly increasing function with a strictly decreasing function? Again, try to prove your answer precisely. 30. Finally, what can you say about the composition of a strictly decreasing function with another strictly decreasing function? 31. What do we mean when we say a function is even? 32. What do we mean when we say a function is odd? 33. Why is the property of a function being even or odd referred to as symmetry? 34. What s a symmetric domain? And how is that related to the concept of even and odd functions? 35. Can a function be simultaneously even and odd? 36. What is the value of an odd function at zero? 37. Is even and odd a dichotomy? (and what s a dichotomy in the first place? Look it up on Wikipedia if necessary.) 38. (Only for students who know derivatives) If f is an even, differentiable function, a. what is the value of f (0)? Justify with a calculation. b. what is the relationship between f (x) and (f( x))? c. Complete the sentence: if a differentiable function is even, then its derivative is. And if a differentiable function is odd, then its derivative is. d. In light of the previous result, can you think of a different way of justifying your answer for part. a? e. What can you say about the value of the first, third, fifth, seventh, etc. derivative of a differentiable, even function at zero? f. What can you say about the value of the zeroth, second, fourth, sixth, etc. derivative of a differentiable, odd function at zero?

4 g. (Only for students who know Taylor approximations) What do the two preceding problems tell you about the Taylor polynomials of an even and odd function? 39. Can an even function be strictly increasing? 40. Can an even function be increasing? 41. How can you tell that a polynomial is even? Odd? 42. Identify each of the six basic trigonometric functions as even or odd. 43. Is the absolute value function even or odd? 44. Integers are also called even and odd, but the meaning is different. (The property of an integer being even or odd is called parity.) Is parity a dichotomy on the integers? 45. Recall the parity rules for multiplication of integers: even times even is even, even times odd is even, odd times odd is odd. Do the same rules hold for multiplication of even and odd functions? Investigate. 46. What about composition of even and odd functions? What are the rules for even composed with even, even composed with odd, odd composed with even, odd composed with odd? 47. Do the rules for composition of even and odd functions follow the template of multiplication of even and odd functions, or multiplication of even and odd numbers, or neither? 48. Can an even function be invertible? Explain. 49. Are all invertible functions odd? If yes, explain. If no, give an example of an invertible function that s not odd. 50. For a function f defined on a symmetric domain, g(x) = f(x)+f( x) is called the symmetrization of f. Which symmetry property does g have? 51. For a function f defined on a symmetric domain, h(x) = f(x) f( x) is called the antisymmetrization or skew-symmetrization of f. Which symmetry property does h 2 2

5 have? 52. What can you say about the symmetrization of an even function? 53. What can you say about the anti-symmetrization of an odd function? 54. How do you find the symmetrization of a polynomial? And the anti-symmetrization? 55. True or false? Any function defined on a symmetric domain can always be broken up into the sum of an even and odd function. Explain. (Extra for students who know power series: think about this problem for the special case of a function given by a power series.) Computational questions: 56. Create a plot on paper of the following function: 57. What are the domain and range of f? 58. Is f increasing? Strictly? x + 1 (x < 0) f(x) = x (0 x < 1) 3 + x 1 (x 1) 59. (Only for students who know derivatives) Compute f (x) and find its domain. What geometric feature does the graph of f have near x=1? 60. Compute (f f)(0). 61. Solve the equation f(x) = Solve the equation (f f)(x) = Verify that f invertible and explain. Then find the inverse function of f and write it again in proper piecewise notation.

6 64. Make a plot of f 1. Verify that it agrees with what you found in the previous question. 65. Is f 1 strictly increasing? Justify. 66. Identify a simple modification of f that makes it continuous.