Chapter 6

Plot graph of the following functions:- i. \(f(x)=x+\frac{1}{x}\). ii. \(f(x)=\ln \left(x^{2}+1\right)\). iii. \(f(x)=\sin ^{-1}(\sin x)\). iv. \(f(x)=\cos ^{-1}(\cos x)\). v. \(\quad f(x)=\tan ^{-1}(\tan x)\).

Plot graph of the following functions:- i. \(y=e^{\ln |x|}\). ii. \(y=x^{\log _{x} 2}\). iii. \(y=\operatorname{sgn}(\sin x)\). iv. \(y=x^{\operatorname{sgn} x}\). v. \(\quad y=\operatorname{sgn}\left(\tan ^{-1} x\right)\). vi. \(y=(\operatorname{sgn} x)^{\operatorname{sgn} x}\). vii. \(y=\sin ^{-1}(\operatorname{sgn} x)\). viii. \(y=\cos ^{-1}(\operatorname{sgn} x)\). ix. \(\quad y=|x-1|+|x-2|+x\). x. \(\quad y=2^{\frac{|x|+x}{x}}\). xi. \(y=\log _{x} x^{2}\). xii. \(y=\ln \tan x+\ln \cot x\).

Plot the graphs of the functions \(\frac{f(x)+|f(x)|}{2}, \frac{f(x)-|f(x)|}{2} \& \frac{|f(x)|}{f(x)}\) from the graph of \(f(x)\) for the following functions:- i. \(f(x)=\ln x\). ii. \(f(x)=x^{3}\). iii. \(f(x)=\sin x\). iv. \(f(x)=\tan x\). v. \(f(x)=\sin ^{-1} x\). vi. \(f(x)=\tan ^{-1} x\).

Plot the graphs of the functions i. \(f(x)=\max \left\\{x, x^{2}\right\\}\). ii. \(f(x)=\min \left\\{x, x^{2}\right\\}\). iii. \(f(x)=\max \\{\sin x, \cos x\\}\). iv. \(f(x)=\min \\{\sin x, \cos x\\}\).

Determine the function \(f(x)=\max \\{(1-x),(1+x), 2\\}\).

Plot the graphs of the functions i. \(f(x)=\max \\{\sin t: 0 \leq t \leq x\\}, \quad x \geq 0\) \(=\max \\{\sin t: x \leq t \leq 0\\}, \quad x<0 .\) ii. \(f(x)=\min \\{\cos t: 0 \leq t \leq x\\}, \quad x \geq 0\) \(=\min \\{\cos t: x \leq t \leq 0\\}, \quad x<0 .\) iii. \(f(x)=\min \\{\sin t: 0 \leq t \leq x\\}, \quad x \geq 0\) \(=\min \\{\sin t: x \leq t \leq 0\\}, \quad x<0\) iv. \(f(x)=\max \\{\cos t: 0 \leq t \leq x\\}, \quad x \geq 0\) \(=\max \\{\cos t: x \leq t \leq 0\\}, \quad x<0\) v. \(\quad f(x)=\max \\{\tan t: 0 \leq t \leq x\\}\). vi. \(f(x)=\min \\{\tan t: 0 \leq t \leq x\\}\).

If \(f(x)=\min \\{|x|,|x-2|, 2-|x-1|\\}\), then draw the graph of \(f(x)\) and also discuss its continuity and differentiability.

Let \(f(x)=x^{3}-x^{2}+x+1\) and \(g(x)=\max \\{f(t): 0 \leq t \leq x\\}, \quad 0 \leq x \leq 1\) \(\quad=3-x, \quad 1

Plot graph of the function \(f(x)=(|x|-1)^{2} e^{\frac{1}{|x|-1}}, \quad x \neq \pm 1\) \(=0, \quad x=\pm 1 .\) Write i. all the points of discontinuity of \(f(x)\); ii. all the points where \(f(x)\) is not differentiable; iii. all the stationary points of \(f(x)\); iv. intervals of monotonicity of \(f(x)\) v. all the critical points of \(f(x)\); vi. all the points of maxima of \(f(x)\); vii. all the points of minima of \(f(x)\); viii. intervals of concavity of \(f(x)\); ix. all the points of inflection of \(f(x)\); x. range of \(f(x)\); xi. greatest and least value of \(f(x)\);

Prove that the function \(y=\frac{k}{x}\) is inverse to itself.

Prove that the function \(y=\frac{1-x}{1+x}\) is inverse to itself.

Prove that the function \(y=\frac{x+2}{x-1}\) is inverse to itself.

Let \(f(x)=\frac{\alpha x}{x+1}, \quad x \neq-1\). For what value of \(\alpha, f(x)\) is the inverse of itself?

Prove that the inverse of the linear-fractional function \(y=\frac{a x+b}{c x+d}(a d-b c \neq 0)\) is also a linear-fractional function. Under what conditions does this function coincides with it's inverse?

Show that the function \(f(x)=\sqrt[n]{a-x^{n}}, x>0\) is inverse to itself.

If \(f(x)=2 x-3\) and \(g(x)=x^{3}+5\), then find \((\text { fog })^{-1}(x)\).

Find the set of all values of \(a\) for which \(f(x)=x^{3}+(a+2) x^{2}+3 a x+5\) is invertible.

If \(f(x)=x^{3}+x+1\), then find \(\left(\frac{d}{d x} f^{-1}(x)\right)_{x=1}\)

If \(f(x)=x+e^{x}\) and \(g(x)\) be its inverse function, then find \(g^{\prime}(1)\).

If \(f(x)=x+\cos x\) and \(g(x)\) be its inverse function, then find \(g^{\prime}\left(\frac{3 \pi}{2}\right)\) and \(g^{\prime}\left(\frac{\pi}{4}+\frac{1}{\sqrt{2}}\right)\).

Show that the function \(f(x)=x^{2}-x+1, \quad x \geq \frac{1}{2}\) and \(g(x)=\frac{1}{2}+\sqrt{x-\frac{3}{4}}\) are mutually inverse, and solve the equation \(x^{2}-x+1=\frac{1}{2}+\sqrt{x-\frac{3}{4}}\)

Find the domain of the function \(f(x)=\frac{1}{\lfloor x-1 \mid]+[|7-x|]-6}\) ([ ] denotes greatest integer function).

\(\lim _{x \rightarrow \infty}\\{x\\}\)

\(\lim _{x \rightarrow 2^{-}} x+(x-[x])^{2} .\)

\(\lim _{x \rightarrow 1} 1-x+[x-1]+[1-x] .\)

\(\lim _{x \rightarrow 0} \frac{x-|x|}{x-[x]}\)

\(\lim _{x \rightarrow 0}\left[\frac{\sin (\operatorname{sgn} x)}{\operatorname{sgn} x}\right]\)

\(\lim _{x \rightarrow 0} x\left[\frac{1}{x}\right]\)

\(\lim _{x \rightarrow \infty} x\left[\frac{1}{x}\right]\)

\(\lim _{x \rightarrow n}(-1)^{[x]}\)

\(\lim _{x \rightarrow 0}\\{x\\}\)

\(\lim _{n \rightarrow \infty} \frac{[n x]}{n}\)

\(\lim _{n \rightarrow \infty} \frac{\left[1^{2} x\right]+\left[2^{2} x\right]+\left[3^{2} x\right]+\ldots \ldots+\left[n^{2} x\right]}{n^{3}}\)

\(\lim _{x \rightarrow 0} \frac{\tan \left(\left[-2 \pi^{2}\right] x^{2}\right)-\left(\tan \left[-2 \pi^{2}\right]\right) x^{2}}{\sin ^{2} x} .\)

\(\lim _{x \rightarrow 0} \lim _{n \rightarrow \infty} \frac{\left[1^{2}(\sin x)^{x}\right]+\left[2^{2}(\sin x)^{x}\right]+\ldots .+\left[n^{2}(\sin x)^{x}\right]}{n^{3}}\)

\(\lim _{x \rightarrow 0}\left(\frac{(1+\\{x\\})^{\frac{1}{\mid x\\}}}}{e}\right)^{\frac{1}{\\{x\\}}}\)

Show that \([x]+\left[x+\frac{1}{2}\right]=[2 x] \forall x\).

Show that \([x]+\left[x+\frac{1}{3}\right]+\left[x+\frac{2}{3}\right]=[3 x] \forall x\).

Show that \(\left[x+\frac{1}{2}\right]-\left[x-\frac{1}{2}\right]=1 \forall x\).

Plot graphs:- i. \(\quad f(x)=[\sin x]\). ii. \(\quad f(x)=\sin ^{-1}[x]\). iii. \(y=\operatorname{sgn}[x]\). iv. \(y=\ln [\sin x]\). v. \(\quad f(x)=\left|e^{\langle x\\}}-2\right|\). vi. \(\quad f(x)=[2 x]-2[x]\). vii. \(f(x)=|[x]|-[|x|]\). viii. \(f(x)=[x]-\left[x-\frac{1}{2}\right]\). ix. \(\quad f(x)=(-1)^{[x]}\). x. \(\quad f(x)=\frac{1}{\\{x\\}}\). xi. \(\quad f(x)=[x]+[-x]\). xii. \(f(x)=\left[x+\frac{1}{2}\right]-\left[x-\frac{1}{2}\right]\).

If \(f(x)=1+x-[x]\), then find the function \(g(x)=\operatorname{sgn}(f(x))\)

Show that the function \(f(x)=[x]+[-x]\) has removable discontinuity for integral values of \(x\).

Prove that \(f(x)=\frac{2 x(\sin x+\tan x)}{2\left[\frac{x+2 \pi}{\pi}\right]-3}\) is an odd function.

Check the function \(\begin{aligned} f(x) &=\frac{\left[x^{2}\right]-1}{x^{2}-1}, \quad x^{2} \neq 1 \\ &=0, \quad x^{2}=1 \end{aligned}\) for continuity at \(x=1\).

For what values of \(a\) and \(b\), the function $$ \begin{aligned} f(x) &=\frac{a+3 \cos x}{x^{2}}, \quad x<0 \\ &=b \tan \frac{\pi}{[x+3]}, \quad x \geq 0 \end{aligned} $$ is continuous at \(x=0\).

Discuss the continuity of the function \(f(x)=[x]+\sqrt{\\{x\\}}, \quad x \geq 0\) \(=\sin x, \quad x<0\)

If \(f(x)=|x-1|\\{x\\}, \quad x \neq 1\) \(=0, \quad x=1 .\) Test the differentiability at \(x=1\).

Draw the graph of the function $$ \begin{aligned} F(x) &=x-[x], & & 2 n \leq x<2 n+1 \\ &=\frac{1}{2} \quad, & & 2 n+1 \leq x<2 n+2 \end{aligned} $$ where \(n\) is an integer. Is the function periodic? If periodic, what is its period? What are the points of discontinuity of \(F(x) ?\\{\)

If \(f(x)=x+[x]\), then find its inverse function.

For what values of the constant \(a\), the function \(f(x)=x+[a x]\) is inverse to itself and plot it's graph.