First slide

Algebra of real valued functions

Question

$\begin{array}{l} f (x) = \{\begin{cases} 0, i f x i s r a t i o n a l \\ x, i f x i s i r r a t i o n a l \end{cases} \\ g (x) = \{\begin{cases} 0, i f x i s i r r a t i o n a l \\ x, i f x i s r a t i o n a l \end{cases} \end{array}$ Then f - g is

Moderate

A

one-one and into
B

neither one-one nor onto
C

many one and onto
D

one-one and onto

Solution

$(f - g) (x) = f (x) - g (x)$ $= \{\begin{matrix} \begin{array}{l} 0 - x = - x, \\ x - 0 = x \end{array} & \begin{array}{l} i f x i s r a t i o n a l \\ i f x i s i r r a t i o n a l \end{array} \end{matrix}$
Clearly (f - g)(x) is one-one and onto.

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