Chapter 1

Define the following analytical functions using basic functions:- i. \(\quad f(x)=\sin x+\ln x, x>1\) ii. \(\quad f(x)=x^{2}-\cos x, x \geq 0\) iii. \(\quad|x|\) iv. \(\operatorname{sgn} x\) v. \(\quad f(x)=e^{x}-\sin x, x<0\) \(=3 \sqrt{x}, \quad x \geq 0\) vi. \(\quad f(x)=\frac{\sin x}{\sqrt{\ln x}+\cosh x}\) vii. \(f(x)=\frac{\sqrt{e^{x}}}{1+x^{2}}, x \leq 0\) \(=\frac{\frac{1}{x}+\ln x}{e^{x}+\ln ^{2} x}, x>0\) viii. \(f(x)=\sqrt{\ln (\sin x)+\sin \sqrt{\ln x}}\) ix. \(\quad f(x)=\left(\frac{x}{1+\sin x}\right)^{3}\) x. \(\quad f(x)=2^{\cos x+\sqrt{x}}\) xi. \(f(x)=x^{x}\) xii. \(f(x)=(\sin x)^{\cos x}\) xiii. \(f(x)=\left(\frac{1+x}{1-x}\right)^{x}\) xiv. \(f(x)=\sin ^{-1}\left(\ln x+\sin ^{-1} x\right)\) xv. \(\quad f(x)=\cos ^{-1}\left(x \ln x+\sqrt{\tan ^{-1} \sqrt{x}}\right)\) xvi. \(f(x)=\log _{\sin x} \cos x\)

Find domain (write conditions only):- i. \(f(x)=\frac{1}{\ln (1-x)}+\sqrt{x+2}\) ii. \(f(x)=\sqrt{\sin ^{-1}\left(\log _{3} x\right)}\) iii. \(\quad f(x)=\sqrt{\ln (\sin x)}+\sin ^{-1}(\sqrt{\ln x})\) iv. \(\quad f(x)=\cos ^{-1}\left(\frac{3}{4+2 \sin x}\right)\) v. \(f(x)=\log _{2}\left(\log _{3} x\right)\) vi. \(\quad f(x)=\frac{x}{\ln \left(1+\sec ^{-1}(\ln x)\right)}\) vii. \(f(x)=\sqrt{4+x}-\sqrt{x+2}+\sqrt{15-x}\) viii. \(f(x)=\frac{1}{\sqrt{\ln \\{\cosh (\sin x)\\}}}\) ix. \(f(x)=\sqrt{-x}+\frac{1}{\sqrt{2+\operatorname{cosec}^{-1}(\sin x)}}\) x. \(\quad f(x)=2^{\frac{1}{\cos ^{-1} x}}+\cos ^{-1}\left(2^{x}\right)\) xi. \(\quad f(x)=\tan \left(\frac{1}{1-\tan ^{-1}\left(e^{x}\right)}\right)\) xii. \(f(x)=\sqrt[3]{\sin x}+\sqrt[4]{\cos x}\) xiii. \(f(x)=x^{x}\) xiv. \(f(x)=(\sin x)^{\cos ^{-1} x}\) xv. \(f(x)=\left(\frac{1+x}{1-x}\right)^{x}\) xvi. \(f(x)=\log _{\sin x} \cos x\)

$$ \begin{aligned} &\text { Find domain of the function }\\\ &\begin{aligned} f(x) &=\frac{1}{x-3}, \quad x>2 \\ &=\frac{1}{x}, \quad-1

$$ \begin{aligned} &\text { If } \begin{aligned} f(x) &=\frac{x^{2}+1}{x-1}, \quad x<3 \\ &=\frac{\sin x}{x-3}, \quad x>3, \end{aligned}\\\ &\text { for what values of } x \text { is the function not defined? \\{Ans. } 1,3\\} \end{aligned} $$

$$ \begin{aligned} &\text { Are the following functions equivalent? }\\\ &\text { i. } \left.\quad f(x)=\frac{x}{x} \text { and } \phi(x)=1 \text { \\{Ans. No }\right\\} \end{aligned} $$ ii. \(\quad f(x)=\frac{1}{\frac{1}{x}}\) and \(\phi(x)=x\) \\{Ans. No\\} iii. \(\quad f(x)=\frac{1}{x}-\frac{1}{x}\) and \(\phi(x)=0\) \\{ns. No\\} iv. \(\quad f(x)=(\sqrt{x})^{2}\) and \(\phi(x)=x\) \\{ Ans. No \(\\}\) v. \(f(x)=\sqrt{x^{2}}\) and \(\phi(x)=x\) \\{Ans. No\\} vi. \(\quad f(x)=\sqrt{x^{2}}\) and \(\phi(x)=|x| \quad\\{\) Ans. Yes \(\\}\) vii. \(f(x)=\log x^{2}\) and \(\phi(x)=2 \log x\) \\{Ans. No\\} viii. \(f(x)=\log (x-2)+\log (x-3)\) and \(\phi(x)=\log (x-2)(x-3)\) \\{ Ans. No ix. \(\quad f(x)=\cot x\) and \(\phi(x)=\frac{1}{\tan x}\) \\{Ans. No \(\\}\) x. \(f(x)=\sin ^{2} x+\cos ^{2} x\) and \(\phi(x)=1\) \\{Ans. Yes \(\\}\) xi. \(\quad f(x)=\frac{1}{\cos e c x}\) and \(\phi(x)=\sin x\) \\{Ans. No\\}

$$ 3^{-\frac{1}{2} \log _{3} 9} \cdot\left\\{\text { Ans. } \frac{1}{3}\right\\} $$

$$ 2^{2-\log _{2} 5} \cdot\left\\{\text { Ans. } \frac{4}{5}\right\\} $$

$$ 10^{\log m+\log n} . $$

$$ 2^{2-\log _{2} 5} $$

$$ (5.8)^{\log _{5.8} 10+1} $$

$$ 8^{\log _{2} \sqrt[3]{121}+\frac{1}{3}} $$

$$ \sqrt{\log _{0.5}^{2} 4} $$

$$ \log _{\pi} \tan (0.25 \pi) $$

$$ \log _{2}\left(\log _{3} 81\right) $$

$$ 81^{\left(\frac{1}{\log _{5} 3}\right)}+27^{\log _{9} 36}+3^{\frac{4}{\log _{7} 9}} $$

$$ 2^{\log _{3} 5}-5^{\log _{3} 2} $$

$$ \log _{3} 5 \cdot \log _{25} 27 $$

$$ \log _{3} 5 \cdot \log _{25} 27 $$

$$ \log _{9} 27-\log _{27} 9 $$

$$ \log _{3} 4 \cdot \log _{4} 5 \cdot \log _{5} 6 \cdot \log _{6} 7 \cdot \log _{7} 8 \cdot \log _{8} 9 $$

$$ \log _{3} 2 \cdot \log _{4} 3 \cdot \log _{5} 4 \cdots \cdots \cdots \log _{15} 14 \cdot \log _{16} 15 $$

$$ \log _{6}(216 \sqrt{6}) $$

$$ \log _{2} \log _{2} \log _{4} 256+2 \log _{\sqrt{2}} 2 $$

$$ \sqrt{\left(\frac{1}{\sqrt{(27)}}\right)^{2-\frac{\left(\log _{5} 13\right)}{2 \log _{5} 9}}} $$

$$ 7 \log \frac{16}{15}+5 \log \frac{25}{24}+3 \log \frac{81}{80} $$

$$ \log _{2} \sqrt[3]{16}+\log _{8} \sqrt[4]{2}-\log _{3}(27 \sqrt{3})-\log _{5} \sqrt{5 \sqrt{5}} $$

$$ \log _{2}\left(\frac{1}{4 \sqrt{4}}\right)+\log _{3}\left(\frac{\sqrt[3]{3 \sqrt{3}}}{27}\right)+\log _{4}\left(\frac{\sqrt[3]{8}}{128 \sqrt{2}}\right)-\log _{7}\left(\frac{\sqrt{7}}{\sqrt[3]{49}}\right) $$

$$ \log _{\frac{1}{3}} \sqrt{9}+\log _{\sqrt[3]{\frac{1}{3}}} 9-\log _{\frac{1}{8}} \sqrt[4]{32}+\log _{\frac{1}{\sqrt{2}}} \sqrt[3]{128 \sqrt{2}} $$

$$ \log _{3} 27-\log _{\sqrt{3}} 27-\log _{\frac{1}{3}} 27-\log _{\frac{\sqrt{3}}{2}}\left(\frac{64}{27}\right) $$

$$ \log _{2}\left(\frac{\sqrt[3]{4} \sqrt{2 \sqrt[5]{16}}}{\sqrt{2}}\right)-\log _{\frac{1}{2}} \sqrt[3]{\frac{4}{\sqrt{2}}}+\log _{\frac{1}{\sqrt{3}}}(9 \sqrt[3]{3}) $$

$$ \log _{0.4}\left(\frac{1}{5} \cdot \sqrt[3]{50}\right)+\log _{0.6}\left(\frac{\sqrt{15}}{5}\right)+\log _{0.32}\left(\frac{2 \sqrt{2}}{5}\right) $$

$$ \log _{\sqrt[5]{5}}^{2} \sqrt{5}-\log _{\sqrt[3]{5}}(5 \sqrt{5})+\log _{(\sqrt{3}+1)}(4+2 \sqrt{3}) . $$

$$ \sqrt{\log _{\sqrt{2}} \sqrt{\sqrt{2} \sqrt{\sqrt{2}}}+\log _{\sqrt{\sqrt{2}}} \sqrt[4]{\sqrt{2 \sqrt{2}}}} $$

$$ \sqrt{\log _{\sqrt{3}} \sqrt[4]{\frac{(\sqrt{3})^{\frac{1}{2}}}{\sqrt{3}}}+\log _{\sqrt[4]{2}} \sqrt[4]{\sqrt{\frac{2}{\sqrt{2}}}}} $$

$$ \left(\log _{\sqrt{5}} \frac{1}{5}\right) \sqrt{\log _{\frac{1}{5}}(5 \sqrt{5})+\log _{\sqrt{5}}(5 \sqrt{5})} $$

$$ 2 \log _{5} \sqrt[4]{5}+\frac{1}{2} \cdot \log _{\sqrt{5}} 25-\log _{5}^{2} \sqrt{5}-2 $$

$$ \frac{1}{2}\left(9^{\log _{25} 5+1}-3^{2\left(\log _{16} 2+\frac{1}{4}\right)}\right)-\log _{\sqrt{2}}(2 \sqrt{2}) $$

$$ \log _{3} \log _{8} \log _{2} 16 $$

$$ \log _{8} \log _{4} \log _{2} 64 $$

$$ \log _{4} \log _{2} \log _{3} 81 $$

$$ \log _{3}\left[\log _{2}^{2}\left(\frac{1}{2}\right)+6 \log _{2} \sqrt{2}+5\right] $$

$$ \left(\log _{\sqrt{5}} 125 \div \log _{5}^{2} 25\right) \cdot\left(\log _{\frac{1}{5}} \sqrt{5} \div \log _{0.2} \sqrt[3]{25}\right) $$

$$ \left[\log _{\frac{1}{2}} \sqrt{\frac{1}{4}}+6 \log _{\frac{1}{4}}\left(\frac{1}{2}\right)-2 \log _{\frac{1}{16}}\left(\frac{1}{4}\right)\right] \div \log _{\sqrt{2}} \sqrt[5]{8} $$

$$ 3^{1+\log _{3} 4}+2^{\log _{2} 3-2} $$

$$ 4^{3+\log _{4} 2}-(1.5)^{\log _{3} 3-1} $$

$$ 2^{3-\log _{4} 3}+7^{2 \log _{7} 2+1} $$

$$ 16^{1-\log _{8} 5}+4^{\frac{1}{2} \log _{2} 3+3 \log _{8} 5} . $$

$$ 9^{2 \log _{3} 2+4 \log _{81} 2} \cdot \sqrt{3^{2+\frac{1}{2} \log _{3} 16}} . $$

$$ \left.(0.1)^{2 \log 0.1-1.5 \log 0.1} \cdot(0.1)^{\left(\log \frac{8}{3}+2-\log 20\right)}\right) $$

$$ 72 \cdot\left(49^{\frac{1}{2} \log _{7} 9-\log _{7} 6}+5^{-\log _{\sqrt{5}}^{4}}\right) $$