Bijective Function

Q. Answer the following by appropriately matching the lists based on the information given in Column I and Column II
​​​​​​$\begin{array}{cc}Column1& Column2\\ \text{a.}f:R\to [\frac{3\pi }{4},\pi )\text{and}f\left(x\right)={\cot }^{-1}(2x-x^2-2),& \\ \text{then}f\text{is}& \text{p. one-one}\\ \text{b.}f:R\to R\text{and}f\left(x\right)=e^{px}\sin qx& \\ \text{where}p,q\in R^{+},\text{then}f\text{is}& \text{q. into}\\ \text{c.}f:R^{+}\to [4,\infty )\text{and}f\left(x\right)=4+3x^2,& \\ \text{then}f\text{is}& \text{r. many-one}\\ \text{d.}f:R\to R\text{and}f\left(f\right(x\left)\right)=x,\forall x\in R& \\ \text{then}f\text{is}& \text{s. onto}\end{array}$

1. $a-p,r;b-q,r;c-p,s;d-q,s$
2. $a-s,r;b-r,s;c-p,q;d-p,s$
3. $a-q,r;b-p,s;c-p,r;d-q,s$
4. $a-p,q;b-q,s;c-r,s;d-p,r$

Q. A function f from the set of natural numbers to integers defined by f(n)=$\{\begin{array}{cc}\frac{n-1}{2},& \text{where n is odd}\\ -\frac{n}{2},& \text{where n is even}\end{array}$ is

1. One-one but not onto
2. Onto but not one-one
3. One-one and onto both
4. Neither one-one nor onto

Q.

f(x) = $a^x$ (where a >1) defined on f: R $\to (0,\infty )$ is -

1. One One into function

2. One one onto function

3. Many one into function

4. Many one onto function

Q. Which of the following functions are bijective?

1. $f:R\to [0,\infty )$ defined as $f\left(x\right)=e^{\text{sgn}\left(x\right)}$
2. $f:[3,4]\to [4,6]$ defined by $f\left(x\right)=|x-1|+|x-2|+|x+3|+|x-4|$
3. $f:R\to R$ defined by $f\left(x\right)=\left[x\right]+\cos \left(\pi \right[x\left]\right)$ where $[.]$ denotes the greatest integer function
4. $f:R\to R$ defined as $f\left(x\right)=min\{x+2,-2x+4\}$

Q. If the following functions have both domain and co-domain as $[-1,1]$, then select those which are not bijective?

1. $\sin \left({\sin }^{-1}x\right)$
2. $\frac{2}{\pi }\sin \left({\sin }^{-1}x\right)$
3. $\text{sgn}\left(x\right)\ln \left(e^x\right)$
4. $\text{sgn}\left(x\right)x^3$

Q.

$Letf:[0,\sqrt{3}]\to [0,\frac{\pi }{3}+lo{g}_{e}2]definedf\left(x\right)=lo{g}_e\sqrt{x^2+1}+ta{n}^{-1}xthenf\left(x\right)is$

1. one – one and onto

2. one – one but not onto

3. onto but not one – one

4. neither one – one nor onto