a) Prove that for any two intervals [a, a + d] and [c, c + d] in R, there exists

a) Prove that for any two intervals [a, a + d] and [c, c + d] in R, there exists a real number t such that f(x) = x + t is a bijection from [a, a + d] and [c, c + d]. (Here a and c can be any real numbers, and d is a positive real number.) b) Prove that for any two intervals [0, b] and [0, c] in R (0 < b and 0 < c), there exists a real number s such that f(x) = sx is a bijection from [0, b] to [0, c]. c) Prove that for any two intervals [a, b] and [c, d] in R (a < b and c < d), there exist real numbers s and t such that f(x) = sx + t is a bijection from [a, b] to [c, d] satisfying f(a) = c and f(b) = d. Hints: Note that the graph of f(x) must be a straight line in all three cases. It may help to start by choosing some specific numbers for a, b, c, d, and then thinking about what you want the graph of f to look like in order to make f(x) a function between the intervals you’re considering. For part c), you can try to solve it directly, or you can try to combine your answers to a) and b). 2. This problem is similar to 14.21. Let f : X → Y be a function between the sets X and Y . Recall that formally, f is defined as a relation f ⊆ X × Y , and f(x) = y if and only if (x, y) ∈ f. Consider the relation R from Y to X defined by (y, x) ∈ R if and only if (x, y) ∈ f (said another way, (y, x) ∈ R ↔ f(x) = y). Prove that R is a function from Y to X if and only if f is a bijection (both one-to-one and onto). Hints: There are several things to prove here, and it may be easiest to think about them separately. For instance, you need to prove that if R is a function from Y to X, then f is 1-1. What is the contrapositive of this statement? Try to write it as explicitly as possible. In order to do this, you will have to work carefully with the precise definitions of function and one-to-one, and you’ll need to think about the negations of these statements. 3. This problem is similar to 15.22. Let c be a real number, and consider the function f : R → R defined by f(x) = |x| + cx. a) For which values of c is the function f(x) one-to-one? Prove your answer. b) For which values of c is f(x) onto? Prove your answer.