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Quiz: Functions

Page history last edited by Terence Tao 9 years, 10 months ago

This quiz is designed to test your knowledge of basic concepts in functions.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. Let Formula be a function. If we say that Formula is "one-to-one", this means that ("One-to-one" is the opposite of "two-to-one" (a two-to-one function can map two different values in Formula to the same value in Formula).)
    1. Every Formula gets mapped to exactly one element in Formula.
      • INCORRECT. Every function Formula has this property (they each map one element to one element, i.e. they are not "one-to-two"). However, this is not what one-to-one means.
    2. For every Formula there is at most one Formula such that Formula.
      • INCORRECT. Every function Formula has this property (they each map one element to one element, i.e. they are not "one-to-two"). However, this is not what one-to-one means.
    3. For every Formula there is at least one Formula such that Formula.
      • INCORRECT. Every function Formula has this property (they each map one element to one element, i.e. they are not "one-to-two" or "one-to-zero"). However, this is not what one-to-one means.
    4. For every Formula there is some Formula such that Formula.
      • INCORRECT. This is what it means for Formula to be onto - which is not the same as one-to-one!
    5. For every Formula there is at most one Formula such that Formula.
      • CORRECT.
    6. For every Formula there is exactly one Formula such that Formula.
      • INCORRECT. This is what it means for Formula to be invertible - which is not the same as one-to-one!
    7. Formula has an inverse Formula.
      • INCORRECT. Invertibility is not the same as one-to-one - to be invertible, one has to be both one-to-one and onto.
  2. Let Formula be a function. What does it mean if we say that Formula is not one-to-one?
    1. There is an element Formula which gets mapped to two different elements Formula.
      • INCORRECT. If Formula is a function, then every element Formula gets mapped to a single element Formula in Formula. The "vertical line test" ensures that an element in Formula cannot be mapped to more than one element of Formula.
    2. There is an element Formula which does not get mapped to anything.
      • INCORRECT. If Formula is a function, then every element Formula gets mapped to a single element Formula in Formula. Since Formula is the domain of Formula, every element Formula will be mapped to something.
    3. There exist two different elements Formula such that Formula.
      • CORRECT.
    4. The kernel (or null space) of Formula is non-zero.
      • PARTIALLY. This is true for certain types of functions (namely, linear transformations). However, for general functions Formula there is no concept of a kernel or null space.
    5. Formula is not invertible.
      • PARTIALLY. It is true that if Formula is not one-to-one, then it cannot be invertible. However, it is possible for a function to be not invertible while still being one-to-one (it could be one-to-one but not onto). So this is not quite what it means for Formula to be not one-to-one.
    6. There is an element Formula which is not mapped to by any element of Formula.
      • INCORRECT. This is what it means for Formula to be not onto, which is different from Formula being not one-to-one.
    7. Formula is onto.
      • INCORRECT. One-to-one and onto are not mutually exclusive.
  3. Let Formula be a function. What does it mean if we say that Formula is onto? (There are two correct answers supplied for this question.)
    1. For each Formula there exists at least one Formula such that Formula.
      • CORRECT.
    2. For each Formula there exists exactly one Formula such that Formula.
      • INCORRECT. This is what it means for Formula to be invertible.
    3. For each Formula there exists at most one Formula such that Formula.
      • INCORRECT. This is what it means for Formula to be one-to-one.
    4. Every element Formula gets mapped to some element in Formula.
      • INCORRECT. This is true for _all_ functions Formula, not just the onto functions!
    5. For every element Formula, the element Formula lies in Formula.
      • INCORRECT. This presumes that Formula is invertible; but one does not need to be invertible in order to be onto.
    6. Formula.
      • INCORRECT. This is true for _all_ functions Formula, not just the onto functions!
    7. Formula.
      • CORRECT.
    8. Formula is one-to-one.
      • INCORRECT. One-to-one and onto are not mutually exclusive.
  4. Let Formula be a function. What does it mean if we say that Formula is not onto?
    1. There exists an element Formula which is not equal to Formula for any Formula.
      • CORRECT.
    2. There exist two elements Formula which map to the same element of Formula.
      • INCORRECT. This is what it means for Formula to not be one-to-one.
    3. The inverse Formula does not exist.
      • PARTIALLY. It is true that if Formula is not onto, then it does not have an inverse, but it is possible to not have an inverse while still being onto (by failing to be one-to-one).
    4. There exists an element Formula which is not mapped to any element in Formula.
      • INCORRECT. This cannot happen because Formula is a function from Formula to Formula.
    5. For every Formula, Formula is not an element of Formula.
      • INCORRECT. This cannot happen because Formula is a function from Formula to Formula.
    6. There exists Formula such that Formula does not lie in Formula.
      • INCORRECT. This assumes that Formula exists, when in fact Formula cannot exist when Formula is not onto.
  5. Let Formula be a function, and let Formula be a subset of Formula. If we say that Formula is an element of Formula, what exactly do we know about Formula?
    1. Formula is an element of Formula.
    2. Formula is an element of Formula.
      • PARTIALLY. This answer is correct if we know that Formula is invertible. However, if Formula is not invertible, then Formula is meaningless, nevertheless we can still talk about Formula and what it means for Formula to belong to Formula.
    3. Formula is an element of Formula.
    4. Formula is equal to Formula for some Formula.
      • CORRECT.
    5. Formula is an element of Formula.
      • PARTIALLY. It is true that since Formula is a subset of Formula and Formula maps Formula to Formula, that Formula is a subset of Formula, so that if we know that Formula is an element of Formula then it must also be an element of Formula. However, it is possible to be an element of Formula without being an element of Formula, so this is only part of the story.
    6. Formula is an element of Formula.
      • INCORRECT. Formula does not necessarily make sense, because Formula lies in Formula, not in Formula, and Formula is only defined on the domain Formula.
    7. Formula.
      • INCORRECT. Formula is a set, and Formula is only an element, so this equation does not make sense.
  6. Let Formula be a function, and let Formula be elements of Formula such that Formula. What do we need about Formula to conclude that Formula is equal to Formula?
    1. Nothing; this is true for all functions Formula.
    2. We need Formula to be one-to-one.
      • CORRECT.
    3. We need Formula to be invertible.
      • PARTIALLY. This is overkill: just being one-to-one will suffice.
    4. We need Formula to be onto.
    5. We need Formula and Formula to lie in Formula.
    6. We need Formula to be continuous.
    7. We need Formula to be always increasing or always decreasing.
      • PARTIALLY. This is enough, but it is possible to conclude Formula from Formula even for functions which are not increasing or decreasing (e.g. Formula).
  7. Let Formula be a function, and let Formula be arbitrary elements of Formula such that Formula. What would we need about Formula to be able to conclude that Formula is equal to Formula?
    1. Nothing; this is true for all functions Formula.
      • CORRECT. This is one of the basic properties of functions: the principle of Substitution. (It is also related to the vertical line test: a single input cannot give two different outputs).
    2. We need Formula to be one-to-one.
    3. We need Formula to be invertible.
    4. We need Formula to be onto.
    5. We need Formula and Formula to lie in Formula.
    6. We need Formula to be continuous.
    7. We need Formula to be a polynomial.
  8. Let Formula be a function, and let Formula be an arbitrary element of Formula. What would we need about Formula to be able to conclude that Formula for some Formula?
    1. Nothing; this is true for all functions Formula.
    2. We need Formula to be one-to-one.
      • INCORRECT. This will ensure that Formula for at most one Formula, but not for at least one Formula.
    3. We need Formula to be invertible.
      • PARTIALLY. This is overkill: just being onto will suffice.
    4. We need Formula to be onto.
      • CORRECT.
    5. We need Formula to lie in Formula.
      • INCORRECT. This presupposes Formula is invertible; but one does not need invertibility to guarantee that Formula takes the form Formula.
    6. We need Formula to obey the Intermediate Value Theorem.
      • INCORRECT. It is true that one can use the intermediate value theorem under some circumstances to find an Formula for which Formula, but there are many situations in which this theorem does not apply and yet one can still conclude that Formula for some Formula.
    7. We need Formula to be differentiable.
  9. Let Formula be a function, and let Formula be an arbitrary element of Formula. What would we need about Formula to be able to conclude that Formula for exactly one Formula?
    1. Nothing; this is true for all functions Formula.
    2. We need Formula to be one-to-one.
      • INCORRECT. This will ensure that Formula for at most one Formula, but not for at least one Formula.
    3. We need Formula to be invertible.
      • CORRECT.
    4. We need Formula to be onto.
      • INCORRECT. This will ensure that Formula for at least one Formula, but not for exactly one Formula.
    5. We need Formula to be continuous.
    6. We need Formula to be differentiable.
    7. We need Formula to be always increasing or always decreasing.
  10. Let Formula be a function, and let Formula be a subset of Formula If we say that Formula is an element of Formula, what exactly do we know about Formula and Formula?
    1. Formula is invertible.
      • INCORRECT. Despite appearances, Formula does not need to be invertible in order for us to talk about the inverse image Formula of a set Formula.
    2. Formula.
      • INCORRECT. Formula is a set, whereas Formula is only an element of that set, so it is nonsensical to try to equate the two.
    3. Formula.
      • INCORRECT. Formula is a set, whereas Formula is only an element of that set, so it is nonsensical to try to equate the two.
    4. Formula is an element of Formula.
      • CORRECT.
    5. Formula for some Formula.
      • PARTIALLY. This answer is correct if Formula is invertible. However, if Formula is not invertible, then Formula is meaningless, nevertheless we can still talk about Formula and what it means for Formula to belong to Formula.
    6. Formula is an element of Formula.
      • PARTIALLY. Since Formula lies in the domain Formula, it is true that Formula is an element of Formula, but this is not the full story, because it is possible to lie in Formula without lying in Formula.
    7. Formula is an element of Formula.
      • INCORRECT. Formula is a subset of Formula, while Formula is a subset of Formula; there is no reason why elements of one should be elements of the other.

 

 

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