Function or Mapping

Study Material (PDF)
Self Assesment
Study Material (PDF)
Self Assesment

Click on each question to see answer & solution.

Question No: #11

Let `f : \mathbb{R} \rarr \mathbb{R}` be a mapping defined by `f(x) = x^3+x`. Prove that `f` is a bijective mapping.

Question No: #12

Let the function `f : \mathbb{R} \rarr \mathbb{R}` be defined by `f(x)=x^3-6`, for all `x \in \mathbb{R}`. Show that `f` is bijective. Also find a formula that defines `f^{-1}(x)`

Question No: #13

Let `f : \mathbb{R} \to \mathbb{R}` be a mapping, defined by `f(x) = x + |x|`, where `x \in \mathbb{R}`. Show that `f` is neither one-one nor onto.

Question No: #14

Let `f : (-1,1) \rarr \mathbb{R}` be a mapping defined by `f(x) = \frac{x}{1-|x|}`. Show that `f^{-1}` exists. Hence find `f^{-1}`.

Question No: #15

Two mappings `f : \mathbb{R} \rarr \mathbb{R}` and `g : \mathbb{R} \rarr \mathbb{R}` are defined by `f(x) = x^2+1` and `g(x) = 3x-2` respectively. Then find the value of `(g \circ f)(-1)`.

Question No: #16

A mapping `f : \mathbb{R} \to \mathbb{R}` defined by $ f(x) = \begin{cases} x, & \text{when } x \in \mathbb{Q} \\ 1-x, & \text{when } x \notin \mathbb{Q} \end{cases}$, where `\mathbb{Q}` is the set of rational number. Show that `f \circ f = I_{\mathbb{R}}`

Question No: #17

Let `\mathbb{R}` be a set of all real numbers and `f : \mathbb{R} \rarr \mathbb{R}` is a mapping defined by `f(x) = 2x+1`. If `(g \circ f)(x)=10x+10`, then find the mapping `g : \mathbb{R} \rarr \mathbb{R}`

Question No: #18

If `f(x)=x^2`, `g(x)=\tanx` and `h(x)=\logx`, then value of `(h \circ (g \circ f))(\sqrt{\frac{\pi}{4}})` is -- (a) `0`, (b) `1`, (c) `\frac{1}{\pi}`, (d) `1/2\log\frac{\pi}{4}`

Question No: #19

Let `\mathbb{R}` be the set of all real numbers and for all `x \in \mathbb{R}`, the mapping `f : \mathbb{R} -> \mathbb{R}` is defined by `f(x)=ax+2`. If `(f \circ f)=I_{\mathbb{R}}`, then find the value of `a`