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

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