Relations and Functions

Two functions f : $\mathrm{R} \to \mathrm{R}$, g : $\mathrm{R} \to \mathrm{R}$ are defined as follows

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{0, \mathrm{x} \mathrm{is} \mathrm{rational} \\ 1, \mathrm{x} \mathrm{is} \mathrm{irational}\right\} \\ \mathrm{g}\left(\mathrm{x}\right) = \left\{- 1, \mathrm{x} \mathrm{is} \mathrm{rational} \\ 0, \mathrm{x} \mathrm{is} \mathrm{irrational}\right\} \end{gathered}$$

Then, (fog) $\left(\mathrm{\pi }\right)$ + (gof) (e) is equal to

• 0

• - 1

• 2

• 1

If the functions f and g are defined by f(x) = 3x - 4, g(x) = 2 + 3x for x $\in$ R respective, then g^-1(f^-1(5)) is equal to

• 1

• $\frac{1}{2}$

• $\frac{1}{3}$

• $\frac{1}{4}$

The least number among $\sqrt[3]{4}, \sqrt[4]{5}, \sqrt[4]{7}, \sqrt[3]{8}$

• $\sqrt[3]{8}$

• $\sqrt[4]{7}$

• $\sqrt[3]{4}$

• $\sqrt[4]{5}$

The function f: $\mathrm{R} \to \mathrm{R}$ is defined by f(x)=3^{- x}. Observe the following statements
of it
I. f is one-one
II. f is onto
III. f is a decreasing function
Out of these, true statement are

• Only I, II

• Only II, III

• Only I, III

• I, II, III

If : R$\to$C is defined by f(x) =e^ix for x ∈ R, then f is (whereC denotes the set of all complex numbers)

• one-one

• onto

• one-one and onto

• neither one-one nor onto

If f(x) = 2x⁴ - 13x² + ax + b is divisible by x² - 3x + 2, then (a, b) is equal to

• (- 9, - 2)

• (6, 4)

• (9, 2)

• (2, 9)

A binary sequence is an array of O's and 1's. The number of n-digit binary sequences which contain even number of 0's is

• 2^{n - 1}

• 2ⁿ - 1

• 2^{n - 1} - 1ⁿ

• 2ⁿ

Let R denote the set of all real numbers and RT denote the set of all positive real numbers. For the subsets A and B of R define f : A $\to$ B by f(x) = x² for x $\in$ A. Observe the two lists given below

If f : R$\to$R² and g : R $\to$R are such that g{f(x)} = $\left|\sin \left(\mathrm{x}\right)\right|$and f{g(x)} = (sin $\left(\sqrt{\mathrm{x}}\right)$)², then a possible choice for f and g is

• f(x) = x², g(x) = $\sin \left(\sqrt{\mathrm{x}}\right)$

• $\mathrm{f}\left(\mathrm{x}\right) = \sin \left(\mathrm{x}\right), \mathrm{g}\left(\mathrm{x}\right) = \left|\mathrm{x}\right|$

• $\mathrm{f}\left(\mathrm{x}\right) = {\sin }^2\left(\mathrm{x}\right), \mathrm{g}\left(\mathrm{x}\right) = \sqrt{\mathrm{x}}$

• $\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^2, \mathrm{g}\left(\mathrm{x}\right) = \sqrt{\mathrm{x}}$

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{Z} \to \mathrm{Z} \mathrm{is} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{\mathrm{x}}{2}, \mathrm{if} \mathrm{x} \mathrm{is} \mathrm{even} \\ 0, \mathrm{if} \mathrm{x} \mathrm{is} \mathrm{odd}\right\}, \mathrm{then} \mathrm{f} \mathrm{is} \end{gathered}$$

• Onto but not one-to-one

• One-to-one but not onto

• One-to-one and onto

• Neither one-to-one nor onto