Question

Let $f\left(x\right)=\{\begin{array}{cc}0& \text{if x is rational}\\ x& \text{if x is irrational}\end{array}$
$g\left(x\right)=\{\begin{array}{cc}0& \text{if x is irrational}\\ x& \text{if x is rational}\end{array}$
Then the function $(f-g)x$ is

• A. odd
• B. even
• C. neither odd nor even
• D. odd as well as even

Answer 1

The correct option is A odd

Let $f-g=h$.
Hence, from the given data
$h\left(x\right)=-x$ if $x$ is rational
$=x$ if $x$ is irrational,
Hence,
$h(-x)=-(-x)=x$ if $x$ is rational
$=-x$ if $x$ is irrational,
Therefore, $h\left(x\right)=-h(-x)$.
Hence, the given function, $f-g$, is an odd function.