Chapter 6

\(f(x)=\sqrt{x-x^{2}}+\sqrt{3 x-x^{2}-2} .\)

\(f(x)=\frac{1+2(x+4)^{-0.5}}{2-(x+4)^{0.5}}\)

\(f(x)=\frac{1}{3-\log _{3}(x-3)}\)

\(f(x)=\frac{\sqrt{x+5}}{\log (9-x)}\)

\(f(x)=\log _{2}\left(\frac{x-2}{x+2}\right)\)

\(f(x)=\sqrt{\log _{12} x^{2}}\)

\(f(x)=\sqrt{\log \frac{3-x}{x}}\)

\(f(x)=\frac{1}{\sqrt{|x|-x}}\)

\(f(x)=\sqrt[4]{x-|x|}+\log (x+2)\)

\(f(x)=\sin ^{-1}\left(\log _{3}\left(\frac{x}{3}\right)\right)\)

\(f(x)=\sin ^{-1}\left(\frac{x-3}{2}\right)-\log (4-x)\)

\(f(x)=\sin ^{-1}(|x-1|-2)\)

\(f(x)=\sin ^{-1} \log _{2}\left(\frac{1}{2} x^{2}\right)\)

\(f(x)=\sqrt{\log _{0.4}\left(\frac{x-1}{x+5}\right)} \times \frac{1}{x^{2}-36}\)

\(f(x)=\log _{\left(\frac{x-2}{x+3}\right)} \sqrt{16-x^{2}}\)

\(f(x)=\cos ^{-1}\left(\frac{2-|x|}{4}\right)+(\log (3-x))^{-1}\)

\(f(x)=\sqrt{\log _{10}\left(\frac{5 x-x^{2}}{4}\right)}\)

\(f(x)=\sqrt{\frac{-\log _{0.3}(x-1)}{-x^{2}+3 x+18}}\)

\(f(x)=\sqrt{\log (\log x)-\log (4-\log x)-\log 3}\)

\(f(x)=\log (\sqrt{x-4}+\sqrt{6-x})\)

\(f(x)=\log \left(\log ^{2} x-5 \log x+6\right)\)

\(f(x)=\log \left(1-\log \left(x^{2}-5 x+16\right)\right)\)

\(f(x)=\log _{0.5}\left\\{-\log _{2}\left(\frac{3 x-1}{3 x+2}\right)\right\\}\)

\(f(x)=\log _{2 x-5}\left(x^{2}-3 x-10\right)\)

Find the set of values of \(x\) for which the function \(f(x)=\frac{1}{x}+2^{\sin ^{-1} x}+\frac{1}{\sqrt{x-2}}\) is defined.

If a function \(f(x)\) is defined for \(x \in[0,1]\), then find the domain of the function \(f(2 x+3)\).

If \(f(x)=x-1, \quad 0 \leq x \leq 2\), find the domain of \(\phi(x)=f(f(x))\)

Plot graph of the following functions and write their range, greatest value and least value:- i. \(f(x)=x \ln x\). ii. \(\quad f(x)=\frac{\ln x}{x}\). iii. \(f(x)=x^{x}\). iv. \(f(x)=x e^{x}\). v. \(f(x)=x e^{-x}\). vi. \(\quad f(x)=e^{\frac{1}{x}}\). vii. \(f(x)=x e^{\frac{1}{x}}\). viii. \(f(x)=x-\ln (x+1)\).

Plot graph:- i. \(\quad f(x)=\| x|-1|\). ii. \(\quad f(x)=|| x-2|-2|\). iii. \(\quad f(x)=(|x|-1)^{2}\). iv. \(\quad f(x)=\mid 1-\frac{1}{|x|}\). v. \(\quad f(x)=|\sin | x-\frac{\pi}{2} \|\). vi. \(\quad f(x)=\sin || x\left|-\frac{\pi}{4}\right|\). vii. \(\quad f(x)=e^{\mid x+1}\). viii. \(\quad f(x)=e^{|x|+1}\). ix. \(\quad f(x)=e^{|x+1|-1}\). x. \(\quad f(x)=e^{\| x-1}\). xi. \(\quad f(x)=\left|e^{|x|}-2\right|\). xii. \(\quad f(x)=\mid \ln (|1-x|)\). xiii. \(\quad f(x)=\ln (1-|x|)\). xiv. \(\quad f(x)=\|\ln |x \|-1|\). xv. \(\quad f(x)=\mid \ln (1-x)\). xvi. \(\quad f(x)=|\ln | x+1 \mid\). xvii. \(\quad f(x)=\ln (|x|+1)\). xviii. \(f(x)=|\ln | x|-1|\). xix. \(\quad f(x)=\operatorname{sgn}(|x|+1)\). xx. \(\quad f(x)=\operatorname{sgn}(|x+1|)\). xxi. \(\quad f(x)=\mid \operatorname{sgn}(|x|-1)\). xxii. \(\quad f(x)=\operatorname{sgn}(|x-1|)\). xxiii. \(f(x)=\| \tanh x\left|-\frac{1}{2}\right|\). xxiv. \(f(x)=\frac{x}{x+1}\). xxv. \(\quad f(x)=\cos ^{-1}(|x|-1)\). xxvi. \(f(x)=|\sqrt{|x|}-1|\). xxvii. \(f(x)=\sqrt{\| x|-1|}\). xxviii. \(f(x)=\frac{1}{2-|x|}\). xxix. \(f(x)=\frac{1}{|x-1|-1}\).

Test the following functions for even, odd or neither:- i. \(\quad f(x)=\log \left(x+\sqrt{1+x^{2}}\right)\). ii. \(\quad f(x)=\log \frac{1-x}{1+x}\). iii. \(f(x)=2 x^{3}-x+1\). iv. \(\quad f(x)=x^{4}-2 x^{2}\). v. \(f(x)=x-x^{2}\). vi. \(\quad f(x)=\sin x-\cos x . vii. \)f(x)=2^{-x^{2}}\(. viii. \)f(x)=\frac{a^{x}+a^{-x}}{2}\(. ix. \)\quad f(x)=\frac{a^{x}-a^{-x}}{2} . x. \(\quad f(x)=\frac{x}{a^{x}-1}\). xi. \(\quad f(x)=2^{x-x^{4}}\). xii. \(\quad f(x)=x \frac{a^{x}+1}{a^{x}-1}\). xiii. \(f(x)=4-2 x^{4}+\sin ^{2} x\). xiv. \(f(x)=\sqrt{1+x+x^{2}}-\sqrt{1-x+x^{2}}\). xv. \(\quad f(x)=\frac{1+a^{k x}}{1-a^{k x}}\). xvi. \(f(x)=\sin x+\cos x\). xvii. \(f(x)=\sqrt[3]{(1-x)^{2}}-\sqrt[3]{(1+x)^{2}}\). xviii. \(f(x)=x^{2}-|x|\). xix. \(\quad f(x)=x \sin ^{2} x-x^{3}\). xx. \(\quad f(x)=\frac{\left(1+2^{x}\right)^{2}}{2^{x}}\). xxi. \(f(x)=\frac{\cos x \sin x}{\tan x+\cot x}\). xxii. \(f(x)=\sin ^{3} x+2 \tan ^{5} x .\) xxiii. \(f(x)=\frac{\sin ^{4} x+\cos ^{4} x}{x+x^{2} \tan x}\). xxiv. \(f(x)=\frac{\sec ^{4} x+\cos e c^{4} x}{x^{3}+x^{4} \cot x}\). xxv. \(f(x)=\frac{x}{e^{x}-1}+\frac{x}{2}+1\).

For what values of \(a\), the function \(f(x)=\left(a^{2}+a-2\right) x+a^{2}+2 a-3\) is (a) even, (b) odd?

Prove that the product of two even or two odd functions is an even function, whereas the product of an even and an odd function is an odd function.

Prove that if the domain of the function \(f(x)\) is symmetrical with respect to \(x=0\), then \(f(x)+f(-x)\) is an even function and \(f(x)-f(-x)\) is an odd function.

Prove that any function \(f(x)\), whose domain is symmetrical about origin, can be presented as a sum of an even and an odd function. Rewrite the following functions in the form of a sum of an even and an odd function:- $$ \text { i. } f(x)=\frac{x+2}{1+x^{2}} \text { . } $$ $$ \text { ii. } f(x)=a^{x} \text { . } $$ $$ \text { iii. } f(x)=x^{2}+3 x+2 \text { . } $$ $$ \text { iv. } f(x)=1-x^{3}-x^{4}-2 x^{5} \text { . } $$ $$ \text { v. } f(x)=\sin 2 x+\cos \frac{x}{2}+\tan x \text { . } $$ $$ \text { vi. } f(x)=(1+x)^{100} \text { . } $$

Extend the function \(f(x)=x^{2}+x\) defined on the interval \([0,3]\) onto the interval \([-3,3]\) in an even and an odd way.

Let the function \(f(x)=x^{2}+x+\sin x-\cos x\) be defined on the interval \([0,1]\). Find the odd and even extensions of \(f(x)\) in the interval \([-1,1]\).

If \(f(x)\) is an odd function and if \(\lim _{x \rightarrow 0} f(x)\) exists, prove that this limit must be zero.

Show that derivative of an even function is an odd function and derivative of an odd function is an even function.

Let \(f(x)\) be an even function and if \(f^{\prime}(0)\) exists, find it's value.

If \(f(x+y)+f(x-y)=2 f(x) f(y) \forall x, y \in R\) and \(f(0) \neq 0\), then determine that \(f(x)\) is an even function or odd function or neither.

Which of the following functions are periodic:- i. \(f(x)=\cos x^{2}\). ii. \(f(x)=x+\sin x\) iii. \(f(x)=\cos \sqrt{x}\).

Find the period of the following functions:- i. \(\quad f(x)=5 \sin 4 x .\) ii. \(\quad f(x)=4 \sin \left(3 x+\frac{\pi}{4}\right)\). iii. \(\quad f(x)=\tan 2 x .\left\\{\right.\) iv. \(\quad f(x)=\cot \frac{x}{2} \cdot\ v. \)f(x)=\sin 2 \pi x . vi. \(\quad f(x)=\sin ^{2} x . vii. \)\quad f(x)=\sin \left(\frac{2 x+3}{6 \pi}\right) . viii. \(f(x)=\sin ^{4} x+\cos ^{4} x .\) ix. \(\quad f(x)=|\cos x| .\) x. \(\quad f(x)=\sin 2 x+\cos 3 x . xi. \)\quad f(x)=3 \sin \frac{x}{2}+4 \cos \frac{x}{2} \cdot\ xii. \(\quad f(x)=\tan ^{-1}(\tan x)\). xiii. \(\quad f(x)=2 \cos \frac{x-\pi}{3} . xiv. \)\quad f(x)=\sin \left(\frac{\pi x}{2}\right)+\cos \left(\frac{\pi x}{2}\right) . xv. \(\quad f(x)=\sin \frac{2 \pi x}{3}+\cos \frac{\pi x}{2}\). xvi. \(\quad f(x)=\sin \frac{\pi x}{3}+\sin \frac{\pi x}{4}\). xvii. \(f(x)=\sin \left(2 \pi x+\frac{\pi}{3}\right)+2 \sin \left(3 \pi x+\frac{\pi}{4}\right)+3 \sin 5 \pi x\). xviii. \(f(x)=\cos (\sin x)\). xix. \(f(x)=\cos (\sin x)+\cos (\cos x)\). xx. \(\quad f(x)=\frac{(1+\sin x)(1+\sec x)}{(1+\cos x)(1+\operatorname{cosec} x)} xxi. \)\quad f(x)=|\sin x|+|\cos x|\(. xxii. \)f(x)=\sin \frac{\pi x}{2}+2 \cos \frac{\pi x}{3}-\tan \frac{\pi x}{4}$.

Plot the graph of a periodic function \(f(x)\) with fundamental period \(T=1\) defined in the interval \((0,1]\) if i. \(f(x)=x\). ii. \(f(x)=x^{2}\). iii. \(f(x)=\ln x\).

Prove that the function \(f(x)=1, \quad x\) is a rational no. \(=0, \quad x\) is a irrational no. is periodic but has no fundamental period.

Prove that if \(f(x)\) is a periodic function with period \(T\), then the function \(f(a x+b)\) is periodic with period \(\frac{T}{|a|}\)

Prove that if the function \(f(x)=\sin x+\cos a x\) is periodic, then \(a\) is a rational number.

Let \(f(x)\) be a function and \(k\) be a positive real number such that \(f(x+k)+f(x)=0 \forall x \in R\). Prove that \(f(x)\) is a periodic function with period \(2 k .\)

Let \(f\) be a real valued function defined for all real numbers \(x\) such that for some fixed \(a>0\) \(f(x+a)=\frac{1}{2}+\sqrt{f(x)-(f(x))^{2}} \forall x\). Show that the function \(f(x)\) is periodic with period \(2 a\).

Let \(f(x)\) be a real valued function with domain \(R\) such that \(f(x+p)=1+\left[2-3 f(x)+3(f(x))^{2}-(f(x))^{3}\right]^{\frac{1}{3}}\) holds good for all \(x \in R\) and some positive constant \(p\), then prove that \(f(x)\) is a periodic function.

For what integral value of \(n\), the function \(f(x)=\cos n x \sin \frac{5 x}{n}\) is periodic with period \(3 \pi\) ?