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Question

# Following question contains statements given in two columns, which have to be matched. The statements in Column-I are labelled as A, B, C and D while the statements in Column - II are labelled as 1, 2, 3, 4. Any given statement in Column -I can have correct matching with ONE OR MORE statement(s) in Column - II.

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Solution

## (A) If putting $$f(x) = 0$$ then we get$$x = 1, 2 .... , 11$$ (many one) and $$f(x)$$ is a polynomial function of degree odd defined from $$R$$ to $$R$$ which is always onto. Hence $$f(x)$$ is many one-onto.(B) $$f(x)\, =\,\displaystyle \frac{5}{(3x\, +\, 4)^{2}}\, >\, 0;\,\ \, x\, \in\, D_{f}$$ (one-one)$$y\, =\,\displaystyle \frac{2x\, +\, 1}{3x\, +\, 4}$$$$x\, =\, \displaystyle \frac{1\, -\, 4y}{3y\, -\, 2}$$$$\Rightarrow\, y\, \neq\, \displaystyle \frac{2}{3}$$$$\therefore$$ Range of f is $$R\, -\, \left \{ \displaystyle \frac{2}{3}\right \}\, \subset$$ co-domain (into)Hence $$f(x)$$ is one-one - into(C) putting $$x = 0$$, $$\pi,\, 2\,\pi$$ .....we get same value of $$f(x)$$ equal to $$2$$ (many - one)$$f(x)\, =\, e^{sin\,x}\, +\, \displaystyle \frac{1}{e^{sin\, x}}\, \Rightarrow\, f(x)\, \geq\, 2\, \, x\, \in\, R$$Range of $$f$$ is $$[2,\, \infty)\, \subset$$ co-domain (into)Hence $$f(x)$$ is many one into(D) $$f(x)\, =\, log\, [(x\, +\, 1)^{2}\, +\, 2]$$at $$x = 0$$ & $$-2$$ we get same value of $$f(x)$$ equal to $$log3$$ (many-one)$$f(x)\, \leq\, log2\, \\ \,x\, \epsilon\, R$$Range of f is $$[log^{2},\, \infty)\, \subset$$ co-domain (into)Hence $$f(x)$$ is many one-into.

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