EXERCISE 1.2 Relations and Functions Questions

1. Show that the function f : ℝ^* → ℝ^* defined by f(x) = 1/x is one-one and onto, where ℝ^* is the set of all non-zero real numbers. Is the result true, if the domain ℝ^* is replaced by ℤ with co-domain being same as ℝ^*?
2. Check the injectivity and surjectivity of the following functions:
   1. f : ℤ⁺ → ℤ⁺ given by f(x) = x²
   2. f : ℤ → ℤ given by f(x) = x²
   3. f : ℝ → ℝ given by f(x) = x²
   4. f : ℤ⁺ → ℤ⁺ given by f(x) = x³
   5. f : ℤ → ℤ given by f(x) = x³
3. Prove that the Greatest Integer Function f : ℝ → ℝ, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
4. Show that the Modulus Function f : ℝ → ℝ, given by f(x) = |x|, is neither one-one nor onto, where |x| is x, if x is positive or 0 and |x| is –x, if x is negative.
5. Show that the Signum Function f : ℝ → ℝ, given by

               f(x) = {
                   1, if x > 0
                   0, if x = 0
                  -1, if x < 0
               }
           

   is neither one-one nor onto.
6. Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.
7. In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
   1. f : ℝ → ℝ defined by f(x) = 3 – 4x
   2. f : ℝ → ℝ defined by f(x) = 1 + x²
8. Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is bijective function.
9. Let f : ℕ → ℕ be defined by f(n) =

               {
                   n/2, if n is odd
                   n/2, if n is even
               }
           

   for all n ∈ ℕ. State whether the function f is bijective. Justify your answer.
10. Let A = ℝ – {3} and B = ℝ – {1}. Consider the function f : A → B defined by

                f(x) = (2x - 3) / (x - 1).
            

    Is f one-one and onto? Justify your answer.
11. Let f : ℝ → ℝ be defined as f(x) = x⁴. Choose the correct answer.
    • (A) f is one-one onto
    • (B) f is many-one onto
    • (C) f is one-one but not onto
    • (D) f is neither one-one nor onto.
12. Let f : ℝ → ℝ be defined as f(x) = 3x. Choose the correct answer.
    • (A) f is one-one onto
    • (B) f is many-one onto
    • (C) f is one-one but not onto
    • (D) f is neither one-one nor onto.