Let the function f: A-> B be defined as f(x) = 2x + 3, where A and B are sets of integers. Determine if the function is bijective (one-to-one and onto).
Answer
Step 4: Given f(a) = b, we can substitute the function definition and solve for a: 2a + 3 = b implies that a = (b - 3) / 2. Since A and B are sets of integers, for every integer b in B, (b - 3) / 2 is an integer in A. Therefore, the function is onto.
Step 1 ：Step 1: Determine if the function is one-to-one (injective). A function is one-to-one if no two different elements in A have the same image in B. In other words, for all x, y in A, if f(x) = f(y), then x = y.
Step 2 ：Step 2: Given f(x) = f(y), we can substitute the function definition: 2x + 3 = 2y + 3. If we solve this equation for x, we find that x = y. Therefore, the function is one-to-one.
Step 3 ：Step 3: Determine if the function is onto (surjective). A function is onto if every element in B has a pre-image in A. In other words, for all b in B, there exists an a in A such that f(a) = b.
Step 4 ：Step 4: Given f(a) = b, we can substitute the function definition and solve for a: 2a + 3 = b implies that a = (b - 3) / 2. Since A and B are sets of integers, for every integer b in B, (b - 3) / 2 is an integer in A. Therefore, the function is onto.
