Chapter 4

Set \(A\) has 3 elements and set \(B\) has 4 elements, the number of injections that can be defined from \(A\) to \(B\) is \(\quad\) [UPSEAT-2001] (a) 144 (b) 12 (c) 24 (d) 64

The number of bijective functions from set \(A\) to itself when \(A\) contains 106 elements is (a) 106 (b) \((106)^{2}\) (c) \(106 !\) (d) \(2^{106}\)

A function \(f\) from the set of natural numbers to integer defined by \(f(n)=\left\\{\begin{array}{cc}\frac{n-1}{2}, & \text { when } n \text { is odd } \\ -n / 2 & \text { when } n \text { is even }\end{array}\right.\) (a) neither one-one nor onto (b) one-one but not onto (c) onto but not one-one (d) one-one and onto both

\(f: R \rightarrow R, f(x)=(x-1)(x-2)(x-3)\) is [Roorkee-1999] (a) one-one but not onto (b) onto but not one-one (c) both one-one and onto (d) neither one-one nor onto

If \(f:[0, \infty) \rightarrow[0, \infty)\) and \(f(x)=\frac{x}{1+x}\), then (a) one-one and onto (b) one-one but not onto (c) onto but not one-one (d) neither one-one nor onto

Let \(E=\\{1,2,3,4\\}\) and \(F=\\{1,2\\}\), then the number of onto functions from \(E\) to \(F\) is [IIT Screening-2001] (a) 14 (b) 16 (c) 12 (d) 8

Let \(f: R \rightarrow R\) be defined as \(f(x)=x|x| .\) Which one of the following is correct? (a) \(f\) is only onto (b) \(f\) is only one-one (c) \(f\) is neither onto nor one-one (d) \(f\) is one-one and onto

Let \(f: N \rightarrow N\) defined by \(f(x)=x^{2}+x+1, x \in\) \(N\), then \(f\) is [AMU-2000] (a) one-one onto (b) Many one onto (c) one-one but not onto (d) None of these

If the functions \(f(x)\) and \(g(x)\) are defined on \(R \rightarrow R\) such that $$ \begin{aligned} &f(x)= \begin{cases}0, & x \in \text { rational } \\ x, & x \in \text { irrational }\end{cases} \\ &g(x)= \begin{cases}0, & x \in \text { irrational } \\ x, & x \in \text { rational }\end{cases} \end{aligned} $$ then \((f-g)(x)\) is (a) one-one and onto (b) neither one-one nor onto (c) one-one but not onto (d) onto but not one-one

Let \(f:(-1,1) \rightarrow B\), be a function defined by \(f(x)=\tan ^{-1} \frac{2 x}{1-x^{2}}\) Then \(f\) is both one-one and onto when \(B\) is the interval (a) \([0, \pi / 2)\) (b) \((0, \pi / 2)\) (c) \((-\pi / 2, \pi / 2)\) (d) \([-\pi / 2, \pi / 2]\)

If a function \(f(x)\) is defined for \(x \in[0,1]\), then the function \(f(2 x+3)\) is defined for (a) \([3 / 2,1]\) (b) \([-3 / 2,-1]\) (c) \([1,-3 / 2]\) (d) \([-1,3 / 2]\)

A condition for a function \(y=f(x)\) to have inverse is that it should be (a) defined for all \(x\) (b) continuous everywhere (c) an even function (d) strictly monotonic and continuous in the domain

The inverse of function \(y=\frac{10^{x}-10^{-x}}{10^{x}+10^{-x}}\) (a) \(y=\frac{1}{2} \log _{10}\left(\frac{1+x}{1-x}\right)\) (b) \(\log _{10} \frac{1-x}{1+x}\) (c) \(2 \log \frac{1+x}{1-x}\) (d) None of these

The inverse of the function \(f(x)=\left\\{1-(x-3)^{4}\right\\}^{1 / 7}\) is (a) \(\left(1-x^{4}\right)^{1 / 7}+3\) (b) \(\left(1-x^{7}\right)^{1 / 4}+3\) (c) \((1-x)^{4 / 7}-3\) (d) None of these

Let \(f(x)=(x+1)^{2}-1,(x \geq-1)\), then set \(s=\left\langle x: f(x)=f^{-1}(x)\right\rangle\) is (a) Blank (b) \(<0,-1>\) (c) \(<0,1,-1>\) (d) \(\left\langle 0,-1, \frac{-3+i \sqrt{3}}{2}, \frac{-3-i \sqrt{3}}{2}\right\rangle\)

If \(f(x)=\frac{2^{x}-2^{-x}}{2^{x}+2^{-x}}\), then \(f^{-1}(x)\) is (a) \(\frac{1}{2} \log _{2} \frac{x}{1-x}\) (b) \(\frac{1}{2} \log _{2} \frac{1+x}{1-x}\) c) \(\frac{1}{2} \log _{2} \frac{1+x}{x}\) (d) \(\frac{1}{2} \log _{2} \frac{2+x}{2-x}\)

Which one of the following function is oneto-one? \(\quad\) [Kerala PET-2008] (a) \(f(x)=\sin x, x \in[-\pi, \pi)\) (b) \(f(x)=\sin x, x \in\left[\frac{-3 \pi}{2}, \frac{-\pi}{4}\right]\) (c) \(f(x)=\cos x, x \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right)\) (d) \(f(x)=\cos x, x \in[\pi, 2 \pi)\)

If \(f(x)=2 x+1\) and \(g(x)=\frac{x-1}{2}\) for all real \(x\) then \((\operatorname{fog})^{-1}\left(\frac{1}{x}\right)\) is equal to \mathrm{\\{} [ K e r a l a ~ P E T - 2 0 0 8 ] ~ (a) \(x\) (b) \(1 / x\) (c) \(-x\) (d) \(-1 / x\)

If \(f: R \rightarrow R\) is defined by \(f(x)=x^{3}\) then \(f^{-1}(8)\) is equal to \mathrm{\\{} [ K a r n a t a k a ~ C E T - 2 0 0 8 ] ~ (a) \(\\{2,-2\\}\) (b) \(\\{2,2\\}\) (c) \(\\{2\\}\) (d) \(\left\\{2,2 \omega, 2 \omega^{2}\right\\}\)