Final answer:
Examples for the given categories are (i) f(x) = 2x + 1 for one-one but not onto, (ii) f(x) = x for one-one and onto, (iii) f(x) = x² for neither one-one nor onto, and (iv) f(x) = x³ for onto but not one-one.
Step-by-step explanation:
Examples of Different Types of Functions
(i) One-one but not onto: An example is the function f(x) = 2x + 1 when considered from the real numbers to the positive real numbers. This function is one-one because each input maps to a unique output, but it is not onto because there are positive real numbers that are not an output of this function (for example, any value between 0 and 1).
(ii) One-one and onto (also known as a bijective function): A basic example is the function f(x) = x from the real numbers to the real numbers. Every real number is the image of exactly one real number, and every real number has a preimage, making the function both one-one and onto.
(iii) Neither one-one nor onto: Consider the function f(x) = x² when mapping real numbers to the interval [0,1]. This function is not one-one because both 1 and -1 map to 1, and it's not onto because no element in the interval maps to negative numbers, for instance, -0.5.
(iv) Onto but not one-one: The function f(x) = x³ where x is an integer and maps to the set of all integers is an example. It is onto because every integer is an output, but not one-one since both 1 and -1, for example, map to 1.