Question

Let two functions $f(x)$ and $g(x)$ are defined on $R \rightarrow R$ such that $f(x)=\begin{cases}x^2, & x \in \text { irrational } \\ 2-x^2, & x \in \text { rational }\end{cases}$ and $g(x)=\begin{cases}2-x^2, & x \in \text { irrational } \\ x^2, & x \in \text { rational }\end{cases}$. Then the function $f+g: R \rightarrow R$ is

• A. injective as well as surjective.
• B. injective but not surjective.
• C. surjective but not injective.
• D. neither surjective nor injective.

Answer 1

Answer: (D)

Clearly $( f + g )( x )= f ( x )+ g ( x )=2 \forall x \in R =$ constant function
Hence the function $( f + g )( x )$ defined on $R \rightarrow R$ is neither surjective nor injective.]