Rational Function

Q. Let $f:(-\infty ,+1]\to R,g:[-1,\infty )\to R$ be such that $f\left(x\right)=\sqrt{1-x}$ and $g\left(x\right)=\sqrt{1+x}$, then $f\left(x\right)+\frac{1}{g\left(x\right)}$ exist if $x\in$

1. $[-1,1]$
2. $(-\infty ,-1)$
3. $(1,\infty )$
4. $(-1,1]$

Q.

$iff:N\to Nisdefinedbyf\left(n\right)=n-(-1{)}^n,then$

1. f is neither one-one nor onto

2. f is onto but not one-one

3. f is one-one but not onto

4. f is both one-one and onto

Q. Let $f:R-\left\{\frac{\alpha }{6}\right\}\to R$ be defined by $f\left(x\right)=\frac{5x+3}{6x-\alpha }.$
Then the value of $\alpha$ for which $\left(fof\right)\left(x\right)=x,$ for all
$x\in R-\left\{\frac{\alpha }{6}\right\},$ is:

1. $8$
2. $6$
3. No such $\alpha$ exists
4. $5$

Q. Let $f:R\to R$ be defined by $f\left(x\right)=2x+|x|$. Then $f\left(2x\right)+f(-x)-f\left(x\right)$ is

1. $2x$
2. $2|x|$
3. $-2x$
4. $-2|x|$

Q. The range of $f\left(x\right)=\frac{x^2+14x+9}{x^2+2x+3},x\in R$ is

1. $R-\{1,3\}$
2. $[-5,4]$
3. $R$
4. $R-\{-5,4\}$

Q. For a function, $y=f\left(x\right)=\frac{x}{1+|x|},x,y\in R$, which among the following is true :

1. $f\left(x\right)$ is a one-one but not an onto function
2. $f\left(x\right)$ is neither a one-one nor an onto function
3. $f\left(x\right)$ is an onto but not a one -one function
4. $f\left(x\right)$ is a one-one and an onto function

Q. If $2|x+1{|}^2-3|x+1|+1=0$, then $x=$

1. $0$
2. $-2$
3. $-\frac{1}{2}$
4. $-\frac{3}{2}$

Q. The number of values of $x$ in the interval $[0,1]$ for which the function $f\left(x\right)=\frac{x+3}{|{\log }_{2}x|-6}$ is not defined are

1. $2$
2. $3$
3. $4$
4. Function exists everywhere $\forall x\in [0,1]$

Q. The domain of the function $\frac{1}{\sqrt{x-|x|}}$ is

1. $\varphi$
2. $(-\infty ,0)$
3. $(0,\infty )$
4. $R-\left\{0\right\}$

Q. If $f\left(x\right)=\frac{x+2}{x^2-8x-4}$, then which of the following is correct?

1. domain is $R-\{4-\sqrt{5},4+\sqrt{5}\}$
2. range is $R$
3. domain is $[4-\sqrt{5},4+\sqrt{5}]$
4. range is $(-\infty ,-\frac{1}{4}]\cup [-\frac{1}{20},\infty )$

Q. The range of $\frac{3x^2+9x+17}{3x^2+9x+7}$ is

1. $[1,41]$
2. $(0,10]$
3. $[11,\infty )$
4. $(1,41]$