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Exercise 1.2
1. Show that the function f : R* → R* defined by {f(x) = \dfrac{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 N with co-domain being same as R*?
2. Check the injectivity and surjectivity of the following functions:
i.
f : N → N given by {f(x) = x^2}
ii.
f : Z → Z given by {f(x) = x^2}
iii.
f : R → R given by {f(x) = x²}
iv.
f : N → N given by {f(x) = x^3}
v.
f : Z → Z given by {f(x) = x^3}
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 = \begin{cases}1, & x \gt 0\\0, & x = 0\\-1, & x \lt 0\end{cases}} 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.
i.
f : R → R defined by {f(x) = 3 – 4x}
ii.
f : R → R defined by {f(x) = 1 + x^2}
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 \begin{cases}\dfrac{n + 1}{2} & ,\ if\ n\ is\ odd\\\dfrac{n}{2} & ,\ if\ n\ is\ even\end{cases} 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) = \left(\dfrac{x - 2}{x - 3}\right)}. Is f one-one and onto? Justify your answer.
11. Let f : R → R be defined as {f(x) = x^4}. 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.
