Engineering Jump Start - Summer 2012 – Mathematics Worksheet #4
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 ?
1
12. True or false? 𝑓 −1 (𝑥) means the same as 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?
Engineering Jump Start - Summer 2012 – Mathematics Worksheet #4
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?
Engineering Jump Start - Summer 2012 – Mathematics Worksheet #4
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?
Engineering Jump Start - Summer 2012 – Mathematics Worksheet #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
51. For a function 𝑓 defined on a symmetric domain, ℎ(𝑥) =
𝑓(𝑥)−𝑓(−𝑥)
is called the anti-
symmetrization of 𝑓. Which symmetry property does g have?
2
2
symmetrization or skew-symmetrization of f. Which symmetry property does ℎ
Engineering Jump Start - Summer 2012 – Mathematics Worksheet #4
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:
𝑥+1
(𝑥 < 0)
2
𝑓(𝑥) = �𝑥 + 2 (0 ≤ 𝑥 < 1)
3 + √𝑥 − 1 (𝑥 ≥ 1)
57. What are the domain and range of 𝑓?
58. Is f increasing? Strictly?
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) = 6.
62. Solve the equation (𝑓 ∘ 𝑓)(𝑥) = 3 + √3.
63. Verify that 𝑓 invertible and explain. Then find the inverse function of 𝑓 and write it
again in proper piecewise notation.
Engineering Jump Start - Summer 2012 – Mathematics Worksheet #4
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.