Chapter 6

For what values of \(a,[x+a]-[x-a]=\operatorname{constan} \forall x\).

If \(f\) be function defined on set of non-negative integers and taking values in the same set. Given that:- i. \(x-f(x)=19\left[\frac{x}{19}\right]-90\left[\frac{f(x)}{90}\right]\) for all non-negative integers; ii. \(1900

\(f(x)=\frac{x}{|x|}\). \\{ns. \(\left.[-1] \cup[1]\right\\}\)

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

\(f(x)=\sqrt{3 x^{2}-4 x+5} .\)

\(f(x)=\log \left(3 x^{2}-4 x+5\right)\)

\(f(x)=\log \left(5 x^{2}-8 x+4\right)\)

\(f(x)=\sqrt{x-1}+2 \sqrt{3-x} .\)

\(f(x)=\log _{2} \frac{\sin x-\cos x+3 \sqrt{2}}{\sqrt{2}}\)

\(f(x)=\sqrt{2-x}+\sqrt{1+x}\)

\(f(x)=\frac{x}{x+1}\)

\(f(x)=\frac{1}{2-\sin 3 x}\)

\(f(x)=\log \sqrt{x^{2}+6 x+10}\)

\(f(x)=\frac{e^{x}-e^{-|x|}}{e^{x}+e^{|x|}}\)

\(f(x)=\frac{1}{\sqrt{4+3 \cos x}}\)

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

\(f(x)=3 \sin \sqrt{\frac{\pi^{2}}{16}-x^{2}}\)

Find domain and range of \(f(x)=\sin \log \left(\frac{\sqrt{4-x^{2}}}{1-x}\right)\).

Let \(A=\\{1,2,3,\\}, B=\\{2,3,4\),\(} then which of the following is function from A\) to \(B:-\) i. \(f_{1}=\\{(1,2),(1,3),(2,3),(3,3)\\} .\) ii. \(f_{2}=\\{(1,3),(2,4)\\}\). iii. \(f_{3}=\\{(1,3),(2,2),(3,3)\\} .\) iv. \(f_{a}=\\{(1,2),(2,3),(3,2),(3,4)\\}\).

Let \(A=\\{-2,-1,0,1,2\\}\) and \(B=\\{0,1,2,3,4,5,6\\}\) and a rule \(f(x)=x^{2}\). Whether \(f: A \rightarrow B\) is a function or not? If yes, find range of \(f\).

Consider a rule \(f(x)=2 x-3\). Whether \(f: N \rightarrow N\) is a function or not?

Let \(A=\\{-2,-1,0,1,2\\}\) and \(f: A \rightarrow I\) given by \(f(x)=x^{2}-2 x-3\). Find range of \(f\). Also find preimages of \(6,-3\) and 5. \\{Ans. \(f(A)=\\{-4,-3,0,5\\} ;\) no pre-image of \(6 ; 0\) and 2 are pre-images of \(-3 ;-2\) is the pre-image of 5\(\\}\)

Given \(A=\\{-1,0,2,5,6,11\\}\) and \(B=\\{-2,-1,0,18,28,108\\}\) and \(f(x)=x^{2}-x-2\). Find \(f(A)\). Whether \(f(A)=B\) or not.

Let \(f: R \rightarrow R\) be given by \(f(x)=x^{2}+3\). Find \(\\{x \mid f(x)=28\\}\). Also find the pre-images of 39 and 2 under \(f\).

Express the following functions as sets of ordered pairs and determine their ranges:- i. \(\quad f: A \rightarrow R, f(x)=x^{2}+1\), where \(A=\\{-1,0,2,4\\}\). ii. \(g: A \rightarrow N, g(x)=2 x\), where \(A=\\{x \mid x \in N, x \leq 5\\}\).

Whether \(g=\\{(1,1),(2,3),(3,5),(4,7)\\}\) a function or not? If \(g\) is defined by the rule \(g(x)=a x+b\), then what values should be assigned to \(a\) and \(b\) ?

If the function \(f\) and \(g\) are given by \(f=\\{(1,2),(3,5),(4,1)\\}\) and \(g=\\{(2,3),(5,1),(1,3)\\} .\) Write fog and gof as set of ordered pairs.

If \(A=\\{1,2,3,4\\}\) and \(B=\\{2,4,6,8\\}\) and \(f: A \rightarrow B\) is given by \(f(x)=2 x\), then write \(f^{-1}\) as a set of ordered pairs.

Let \(f(x)=x^{3}\) be a function with domain \(\\{0,1,2,3\\}\). Find the domain of \(f^{-1}\).

Which of the following functions are one-one, many-one, into, onto \& bijective:- i. \(f: R \rightarrow R, f(x)=\sin x . ii. \)f: R \rightarrow[-1,1] f(x)=\sin x\(. iii. \)f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow R, f(x)=\sin x .\( iv. \)f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow[-1,1], f(x)=\sin x .\( v. \)f:[0, \pi] \rightarrow[-1,1], f(x)=\cos x\(. vi. \)f: R^{+} \rightarrow R, f(x)=2 \sqrt{x}+1\(. vii. \)f: R \rightarrow R, f(x)=x^{3}+3\(. viii. \)f(x)=\ln x\(. ix. \)f:(0,1) \rightarrow R, f(x)=\ln x . x. \(\quad f: R \rightarrow I, f(x)=[x]\). xi. \(f: N \rightarrow N, f(x)=[x]\). xii. \(f: R \rightarrow I, f(x)=\operatorname{sgn} x . xiii. \)f: R \rightarrow[0,1), f(x)=\\{x\\}\(. xiv. \)f:[0,1) \rightarrow[0,1), f(x)=\\{x\\}\(. xv. \)f: N \rightarrow N, f(x)=2 x+3$.

Which of the following are functions? i. \(\left\\{(x, y): y^{2}=x, x, y \in R\right\\}\). ii. \(\\{(x, y): y=|x|, x, y \in R\\}\). \\{Ans. function\\} iii. \(\left\\{(x, y): x^{2}+y^{2}=1, x, y \in R\right\\}\). iv. \(\left\\{(x, y): x^{2}-y^{2}=1, x, y \in R\right\\}\). v. \(\left\\{(x, y) \mid x, y \in R, x^{2}=y\right\\}\). vi. \(\left\\{(x, y) \mid x, y \in R, y^{2}=x\right\\}\). vii. \(\left\\{(x, y) \mid x, y \in R, x=y^{3}\right\\}\). viii. \(\left\\{(x, y) \mid x, y \in R, y=x^{3}\right\\}\).

If a function \(f:[2, \infty) \rightarrow B\) defined by \(f(x)=x^{2}-4 x+5\) is a bijection, then find \(B\).

Find the minimum value of \(\frac{\left(x+\frac{1}{x}\right)^{6}-\left(x^{6}+\frac{1}{x^{6}}\right)-2}{\left(x+\frac{1}{x}\right)^{3}+x^{3}+\frac{1}{x^{3}}}\) for \(x>0\).

If \(f(x)=x\) and \(g(x)=|x|\), then find the function \(\phi(x)\) satisfying \([\phi(x)-f(x)]^{2}+[\phi(x)-g(x)]^{2}=0\).

If \(f(x)=\left(a x^{2}+b\right)^{3}\), then find the function \(g\) such that \(f(g(x))=g(f(x))\).

Draw a graph of the function \(f(x)=x-\left|x-x^{2}\right|, \quad-1 \leq x \leq 1\) and discuss its continuity or discontinuity in the interval \(-1 \leq x \leq 1\).

The function \(f\) is defined by \(y=f(x)\) where \(x=2 t-|t|, y=t^{2}+t|t|, t \in R\). Draw the graph of \(f\) for the interval \(-1 \leq x \leq 1\). Also discuss it's continuity and differentiability at \(x=0\).

Given \(a f(x)+b f\left(\frac{1}{x}\right)=\frac{1}{x}-5, x \neq 0, a \neq b\). Find \(f(x)\).