Chapter 3

The domain of the definition of the function \(y=\frac{1}{\log _{10}(1-x)}+\sqrt{x+2}\) [IIT-83; Roorkee-93] (a) \([-2,1)-[0]\) (b) \([2,-1]\) (c) \([-1,2]\) (d) None of these

Domain of \(y=\sqrt{1+x}+\sqrt{2-x}\) (a) \([1,2],[0, \sqrt{6}]\) (b) \([-1,2],[0, \sqrt{3}]\) (c) \([-1,2] ;[\sqrt{3}, \sqrt{6}]\) (d) None of these

Domain of \(f(x)=\frac{1}{x}+2^{\sin ^{-1} x}+\frac{1}{\sqrt{x-2}}\) (a) \([1,2]\) (b) \([2,1]\) (c) \([0,2]\) (d) \(\phi\)

Domain of the function \(f(x)=\frac{x-3}{(x-1) \sqrt{x^{2}-4}}\) (a) \((1,2)\) (b) \((-\infty,-2) \cup(2, \infty)\) (c) \((-\infty,-2) \cup(1, \infty)\) (d) \((-\infty, \infty)-(1, \pm 2)\)

For what value of \(x\) function be identical \(f(x)=\log (x-1)-\log (x-2)\) and \(g(x)=\) \(\log \left(\frac{x-1}{x-2}\right)\) (a) \((1, \infty)\) (b) \((2, \infty)\) (c) \((3, \infty)\) (d) None of these

If \(f(x)=\log \frac{1+x}{1-x}\), then \(f(x)\) is (a) Even Function (b) \(f\left(x_{1}\right) f\left(x_{2}\right)=f\left(x_{1}+x_{2}\right)\) (c) \(\frac{f\left(x_{1}\right)}{f\left(x_{2}\right)}=f\left(x_{1}-x_{2}\right)\) (d) Odd Function

Which of the following is an odd function (a) \(f(x)=\cos x\) (b) \(y=2^{-x^{2}}\) (c) \(y=2^{x-x^{2}}\) (d) None of these

Which of the following function is even function (a) \(f(x)=\frac{a^{x}+1}{a^{x}-1}\) (b) \(f(x)=x\left(\frac{a^{x}-1}{a^{x}+1}\right)\) (c) \(f(x)=\frac{a^{x}-a^{-x}}{a^{x}+a^{-x}}\) (d) \(f(x)=\sin x\)

If the real valued function \(f(x)=\frac{a^{x}-1}{x^{n}\left(a^{x}+1\right)}\), is even then \(n\) equals (a) 2 (b) \(2 / 3\) (c) \(1 / 4\) (d) \(-1 / 3\)

Period of \(\sin \left(2 \pi x+\frac{\pi}{3}\right)+2 \sin \left(3 \pi x+\frac{\pi}{4}\right)+\) \(3 \sin 5 \pi x\) is (a) 1 (b) \(2 / 3\) (c) \(2 / 5\) (d) 2

The period of \(f(x)=\left|\sin ^{3} \frac{x}{2}\right|\) is (a) \(\pi\) (b) \(2 \pi\) (c) \(3 \pi\) (d) None of these

The value of \(n \in I\), for which the function \(f(x)=\frac{\sin n x}{\sin \frac{x}{n}}\) has \(4 \pi\) as its period is \(n\) is equal to (a) 2 (b) 3 (c) 5 (d) 4

Period of \(f(x)=\sin \frac{\pi}{2} x+2 \cos \frac{\pi}{3} x-\tan \frac{\pi}{4} x\) is equal to (a) 4 (b) 8 (c) 12 (d) 16

The function \(f(x)=\frac{1}{2}\left\\{\frac{|\sin x|}{\cos x}+\frac{\sin x}{|\cos x|}\right\\}\) is periodic with period (a) \(\pi\) (b) \(\pi / 2\) (c) \(2 \pi\) (d) \(3 \pi\)

Period of the function \(y=\sin \frac{2 t+3}{6 \pi}\) is (a) \(3 \pi^{2}\) (b) \(5 \pi^{2}\) (c) \(7 \pi^{2}\) (d) \(6 \pi^{2}\)

If \(g(x)=1+\sqrt{x}\) and \(f(g(x))=3+2 \sqrt{x}+x\) then \(f(x)\) is equal to (a) \(1+2 x^{2}\) (b) \(2+x^{2}\) (c) \(1+x\) (d) \(2+x\)

If \(f(x)=\frac{x}{\sqrt{1+x^{2}}}\), then \(f o f \circ f(x)\) is equal to [RPET-2000] (a) \(\frac{3 x}{\sqrt{1+x^{2}}}\) (b) \(\frac{x^{3}}{\sqrt{1+x^{6}}}\) (c) \(\frac{x}{\sqrt{1+3 x^{2}}}\) (d) None of these

If \(g(x)=x^{2}+x-2\) and \(\frac{1}{2}(\) gof \()(x)=2 x^{2}-5 x+2\) then \(f(x)\) is equal to (a) \(2 x-3\) (b) \(2 x+3\) (c) \(2 x^{2}+3 x+1\) (d) \(2 x^{2}-3 x-1\)

Domain of definition of the function \(f(x)=\frac{3}{4-x^{2}}+\log _{10}\left(x^{3}-x\right)\) is (a) \((-1,0) \cup(1,2) \cup(2, \infty)\) (b) \((1,2)\) (c) \((-1,0) \cup(1,2)\) (d) \((1,2) \cup(2, \infty)\)

If \(f(x)=x\) and \(g(x)=|x|\), then what is \((f+g)\) \((x)\) equal to? \(\quad\) INDA-2008] (a) 0 for all \(x \in R\) (b) \(2 x\) for all \(x \in R\) (c) \(\left\\{\begin{array}{c}2 x \text { for } x \geq 0 \\ 0 \text { for } x<0\end{array}\right.\) (d) \(\left\\{\begin{array}{c}0 \text { for } x \geq 0 \\ 2 x \text { for } x<0\end{array}\right.\)

The period of the function \(f(x)=a^{\langle\tan (\pi x)+x-[x]\\}}\), where \(a>0,[.]\) denotes the greatest integer function and \(x\) is a real number, is [Kerala PET-2007] (a) \(\pi\) (b) \(\pi / 2\) (c) \(\pi / 4\) (d) 1

If \(f(x)=e^{x}\) and \(g(x)=\log _{e} x\) then which of the following is true [MPPET-2008] (a) \(f\\{g(x)\\} \neq g\\{f(x)\\}\) (b) \(f\\{g(x)\\}=g\\{f(x)\\}\) (c) \(f\\{g(x)\\}+g\\{f(x)\\}=0\) (d) \(f\\{g(x)\\}-g\\{f(x)\\}=1\)

The domain of the real valued function \(f(x)=\sqrt{1-2 x}+2 \sin ^{-1}\left(\frac{3 x-1}{2}\right)\) is \mathrm{\\{} \text {\\{} \text {\\{} K e r a l a ~ P E T - 2 0 0 7 ] ~ (a) \(\left[\frac{-1}{3}, 1\right]\) (b) \(\left[\frac{1}{2}, 1\right]\) (c) \(\left[\frac{-1}{2}, \frac{1}{3}\right]\) (d) \(\left[\frac{-1}{3}, \frac{1}{2}\right]\)