Using Graph to Find Range of a Function

Q. Fot the function $f\left(x\right)=\frac{e^x+1}{e^x-1}$, If $n\left(d\right)$ denotes the number of integers which are not in its domain and $n\left(r\right)$ denotes the number of integers which are not in its range, then $n\left(d\right)+n\left(r\right)$ is equal to

Q.

If ${\left(3\right)}^{x+y}=81and{\left(81\right)}^{x-y}=3$, find the values of $xandy$

Q.

$1+\frac{3}{2}+\frac{5}{2^2}+\frac{7}{2^3}+...\infty isequalto$

1. $3$

2. $6$

3. $9$

4. $12$

Q. The function $f$ is defined by $f\left(x\right)=\{\begin{array}{cc}1-x,& x<0\\ 1,& x=0\\ x+1,& x>0\end{array}$ Draw the graph of $f\left(x\right)$.

Q.

If ${\mathrm{z}}_1$ and ${\mathrm{z}}_2$ are two complex numbers such that $\left|{\mathrm{z}}_1\right|=\left|{\mathrm{z}}_2\right|+\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|,$ then

1. $\mathrm{Im}\left(\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}\right)=0$

2. $\mathrm{Re}\left(\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}\right)=0$

3. $\mathrm{Re}\left(\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}\right)=0=\mathrm{Im}\left(\frac{{\mathrm{z}}_1}{{\mathrm{z}}_2}\right)$

4. None of these

Q. The range of $f\left(x\right)=\frac{3}{5+4sin3x}$ is

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

Q. If $f:R\to R$ is defined by $f\left(x\right)=\frac{sin\left[x\right]\pi +tan\left[x\right]\pi }{1+\left[x^2\right]}$, then the range of f(x) (where [x] denotes integral part of x)

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

Q. The range of f(x) = |x – 2| + |x – 12| is

1. $[2,\infty )$
2. $[12,\infty )$
3. $[10,\infty )$
4. $[14,\infty )$

Q.

Define an identity function and draw its graph, also find its domain and range.

Q. If $f:R\to R$, then $f\left(x\right)=(x-1)(x-2)(x-3)$ is

1. surjective but not injective
2. injective but not surjective
3. both injective and surjective
4. neither injective nor surjective

Q. $\text{Let}f\left(x\right)=\frac{x-\left[x\right]}{1+x-\left[x\right]},xϵR,\left[\right]\text{dentoes the greatest integer function}.\text{Then, the range of f is}$

1. 

2. 

3. (0, 1)

4. [0, 1]

Q. The range of $f\left(x\right)=\left[x\right]-x.$ Where $\left[x\right]$ is the greatest integer function.

1. $(-1,0]$
2. $(-1,1]$
3. $(1,0)$
4. $(1,2]$

Q. Let $f\left(x\right)=\frac{x-\left[x\right]}{1+x-\left[x\right]},xϵR$, where [ x] denotes the greatest integer function. Then, the range of f is

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

Q. Let $f\left(x\right)=\frac{x-\left[x\right]}{1+x-\left[x\right]},xϵR$, where [ x] denotes the greatest integer function. Then, the range of f is

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

Q. If $x,y,z$ are real numbers such that $x+y+z=4$ and $x^2+y^2+z^2=6,$ then the range of $x$ is

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

Q. The range of the function $f\left(x\right)=\frac{x+3}{|x+3|},x\ne -3$ is

1. $\{3,-3\}$
2. All positive integers
3. $R-\{-3\}$
4. $\{-1,1\}$

Q.

The function f is defined by
$f\left(x\right)=\{\begin{array}{cc}1-x,& x<0\\ 1,& x=0\\ x+1,& x>0\end{array}$ Draw the graph of f(x).

Q. The range of $|x-2|+|x-5|$ is

1. $[2,\infty )$
2. $[3,\infty )$
3. $[4,\infty )$
4. $[5,\infty )$

Q. The set of value(s) of $b$ for which function $f\left(x\right)=\{\begin{array}{cc}{x}^2+b^2-5b+6,& x<0\\ x,& x\ge 0\end{array}$ has a point of minima at $x=0,$ is

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

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

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

Q.

The function $f\left(x\right)=2\left|x\right|+|x+2|-||x+2|-2\left|x\right||$
has a local minimum or a local maximum at x=

1. -2

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

3. 2

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

Q. Consider the functions $f\left(x\right)=\{\begin{array}{c}x+1,x\le 1\\ 2x+1,1<x\le 2\end{array}$

$g\left(x\right)=\{\begin{array}{c}{x}^2,-1\le x<2\\ x+2,2\le x\le 3\end{array}$

Range of the function $f\left(g\right(x\left)\right)$ is

1. $[1,5]$
2. $[2,3]$
3. $[1,2]\cup (3,5]$
4. $[1,5]-\left\{3\right\}$

Q. If $R$ is the set of all real numbers and if $f:R-\left\{2\right\}\to R$ is defined by $f\left(x\right)=\frac{2+x}{2-x}$ for $x\in R-\left\{2\right\}$, then the range of $f$ is

1. $R$
2. $R-\{-2\}$
3. $R-\{-1\}$
4. $R-\left\{1\right\}$

Q. The range of the function $f\left(x\right)=\frac{x+3}{|x+3|},x\ne -3$ is

1. $\{3,-3\}$
2. $R-\{-3\}$
3. All positive integers
4. $\{-1,1\}$

Q. Let $f\left(x\right)={\sin }^{-1}x+2{\tan }^{-1}x+x^5-5x^4+5x^3+1$. Then

1. maximum value of $f\left(x\right)$ is $\pi +2$
2. minimum value of $f\left(x\right)$ is $-\frac{3\pi }{2}-6$
3. maximum value of $f\left(x\right)$ is $-\frac{3\pi }{2}+6$
4. minimum value of $f\left(x\right)$ is $-\pi -10$

Q. If $f:R\to R$ is defined by $f\left(x\right)=\frac{sin\left[x\right]\pi +tan\left[x\right]\pi }{1+\left[x^2\right]}$, then the range of f(x) (where [x] denotes integral part of x)

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

Q.

Find the maximum and minimum values, if any, of the following function given by,

$h\left(x\right)=x+1,xϵ(-1,1)$

Q. Range of the function $f\left(x\right)=\frac{1}{3|x|+2}$ is

1. $R$
2. $(0,\frac{1}{2}]$
3. $[\frac{1}{2},\infty )$
4. $R^{+}$

Q.

Find the range of the function given in graph

1. R

2. None of these

3. (-∞, 1]

4. (-∞, 0) U 1

Q. The range of $|x-2|+|x-5|$ is

1. $[2,\infty )$
2. $[3,\infty )$
3. $[4,\infty )$
4. $[5,\infty )$