Using Monotonicity to Find the Range of a Function

Q.

If $f\left(x\right)=\frac{\left[g\right(x)+g(-x\left)\right]}{2}+\frac{2}{{\left[h\right(x)+h(-x\left)\right]}^{-1}}$, where $g$ and $h$ are differentiable functions, then $f\left(0\right)$is

1. $1$

2. $\frac{1}{2}$

3. $\frac{3}{2}$

4. $0$

Q. The range of the function f(x) = sin [x], $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ where [x] denotes the greatest integer, is

1. {0}
2. {0, -1}
3. {0, $\pm$ sin 1}
4. {0, - sin 1}

Q. If $m=-2,$ then find the value of
$m^2+6m.$

1. 8
2. 9
3. - 8
4. - 9

Q.

For real $x$, let $f\left(x\right)=x^3+5x+1$, then

1. f is onto R but not one-one
   

2. f is one-one and onto R
   

3. f is neither one-one nor onto R
   

4. f is one-one but not onto R

Q. Which of the following functions are into functions if they are all defined from $R$ to $R$?

1. $f\left(x\right)=e^x$
2. $f\left(x\right)=\text{cos}x$
3. $f\left(x\right)=x^3$
4. $f\left(x\right)=\text{sin}x$

Q. Let $f\left(x\right)=(1+b^2)x^2+2bx+1$ and m(b) the minimum value of f(x) for a given b. As b varies, the range of m(b) is

1. $[0,1]$
2. $(0,\frac{1}{2}]$
3. $[\frac{1}{2},1]$
4. $(0,1]$

Q.

The range of $f\left(x\right)=ta{n}^{-1}(x^2+x+a)$ $\forall xϵ$ R is a subset of $[0,\frac{\pi }{2})$ then the range of a is -

1. $[\frac{1}{4},\infty )$
2. $(\frac{-\pi }{2},\frac{\pi }{2})$
3. $[-\sqrt{3},\frac{1}{4}]$
4. $[-\sqrt{3},-1]$

Q.

$f:R\to R$ is defined as $f\left(x\right)=x^4-6x^2+12$. The range of f(x) is

1. $(-\infty ,\infty )$

2. $[12,\infty )$

3. $[-3,\infty )$

4. $[3,\infty )$

Q. Let $f:R\to \left[0.\frac{\pi }{2}\right)$ be defined by $f\left(x\right)=ta{n}^{-1}(x^2+x+a)$. Then the set of values of a for which f is onto is

1. $[0,\infty )$
   
2. $[\frac{1}{4},\infty )$
   
3. $(-\infty ,\frac{1}{4}]$
   
4. $\left\{\frac{1}{4}\right\}$

Q.

Let $f,g$ and $h$be real valued functions defined on the interval [0, 1] by$f\left(x\right)=e^{x^2}+e^{-x^2},g\left(x\right)=x{e}^{x^2}+e^{-x^2}$ and $h\left(x\right)=x^2{e}^{x^2}+e^{-x^2}$. If a, b and c denote, respectively, the absolute maximum of f, g and h on [0, 1], then

1. a = v and c $\ne$ b

2. a $\ne$ b and c$\ne$ b

3. a = c and a $\ne$ b

4. a = b = c

Q. $f\left(x\right)=x^9+3x^7+6$ is increasing in

1. $(-\infty ,\infty )$
2. $(-\infty ,0)$
3. $(0,\infty )$
4. $\varnothing$

Q.

Range of $f\left(x\right)=\frac{tan\left(\pi \right[x^2-x\left]\right)}{1+sin\left(cosx\right)}$ is (where [x] denotes the greatest integer function)

1. $(-\infty ,\infty )\sim [0,tan1]$
2. $(-\infty ,\infty )\sim [tan2,0]$
3. $[tan2,tan1]$
4. $\left\{0\right\}$

Q. The range of $si{n}^{-1}x-co{s}^{-1}x$ is

1. $[\frac{-3\pi }{2},\frac{\pi }{2}]$
2. $[\frac{-5\pi }{2},\frac{\pi }{3}]$
3. $[\frac{-3\pi }{2},\pi ]$
4. $[0,\frac{\pi }{2}]$

Q. Range of the function $f\left(x\right)=\frac{x^2+x+2}{x^2+x+1};xϵR$ is

1. $(1,\infty )$
2. $(1,\frac{11}{7}]$
3. $(1,\frac{7}{3}]$
4. $[1,\frac{7}{5}]$

Q. The range of $\frac{x^2-x+1}{x^2+x+1}$ is

1. $[\frac{1}{3},3]$
2. $[\frac{1}{3},1]$
3. $[1,3]$
4. $(-\infty ,\frac{1}{3}]\cup [3,\infty )$

Q. The range of the function $f\left(x\right)=co{s}^2\frac{x}{4}+sin\frac{x}{4},xϵR$ is

1. $[0,\frac{5}{4}]$
2. $[1,\frac{5}{4}]$
3. $(-1,\frac{5}{4})$
4. $[-1,\frac{5}{4}]$