Question

Let $f:R\to R$ and $g:R\to R$ be two bijective functions such that they are the mirror images of each other about the line $y=a.$ If $h:R\to R$ given by $h\left(x\right)=f\left(x\right)+g\left(x\right),$ then $h\left(x\right)$ is

• A. a one-one and an onto function
• B. a one-one but not an onto function
• C. an onto but not a one-one function
• D. Neither a one-one nor an onto function

Answer 1

The correct option is D Neither a one-one nor an onto function

$y=f\left(x\right)$ and $y=g\left(x\right)$ are mirror image of each other about line $y=a$


Let for some $x=b,g\left(b\right)-a=a-f\left(b\right)$
$\Rightarrow f\left(b\right)+g\left(b\right)=2a$
So, we have $h\left(b\right)=f\left(b\right)+g\left(b\right)=2a$ (constant)
Hence $h\left(x\right)$ is a constant function.
Thus $h:R\to R$ is neither a one-one nor an onto function.