Course Content

Lecture – 2

01:08:07

Questions Discussed:

Question 6. the injectivity and surjectivity of the following functions: (N)
  a. f:\mathbf{N}\to \mathbf{N} given by f(x)={{x}^{2}}
  b. f:\mathbf{Z}\to \mathbf{Z} given by f(x)={{x}^{2}}
  c. f:\mathbf{R}\to \mathbf{R} given by f(x)={{x}^{2}}
  d. f:\mathbf{N}\to \mathbf{N} given by f(x)={{x}^{3}}
  e. f:\mathbf{Z}\to \mathbf{Z} given by f(x)={{x}^{3}}

Question 7.Prove that the Greatest Integer Function f:\mathbf{R}\to \mathbf{R}, given by f(x)=[x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x. (N)

Question 8.Show that the Modulus Function f:\mathbf{R}\to \mathbf{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. (N)

Question 9.Show that the Signum Function f:\mathbf{R}\to \mathbf{R}, given by {\displaystyle f(x)=\begin{cases} 1, & \text{ if } x > 0 \\ 0, & \text{ if } x=0 \\ -1, & \text{ if } x < 0 \end{cases} } is neither one-one nor onto. (N)

Question 10.In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (N)
  a. f:\mathbf{R}\to \mathbf{R} defined by f(x)=3-4x
  b. f:\mathbf{R}\to \mathbf{R} defined by f(x)=1+{{x}^{2}}

Question 11. Let f:\mathbf{R}\to \mathbf{R} be defined as f(x)={{x}^{4}}. Choose the correct answer. (N)
  a.f is one-one onto
  b.f is one-one but not onto
  c.f is many-one onto
  d.f is neither one-one nor onto.

Question 12.Let f:\mathbf{R}\to \mathbf{R} be defined as f(x)=3x. Choose the correct answer.(N)
  a.f is one-one onto
  b.f is one-one but not onto
  c.f is many-one onto
  d.f is neither one-one nor onto.