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