Question

If $f$ is an increasing and $g$ is a decreasing function and $fog$ is defined, then $fog$ will be

• A. increasing function
• B. decreasing funtion
• C. neither increasing nor decreasing
• D. None of these

Answer 1

The correct option is B decreasing funtion

Let $h\left(x\right)=f\left(g\right(x\left)\right)$
Hence
$h^{\prime }\left(x\right)$ $=f^{\prime }\left(g\right(x\left)\right).g^{\prime }\left(x\right)$
Now $g\left(x\right)$ is a strictly decreasing function.
Then
$g^{\prime }\left(x\right)<0$ for all real $x.$
Hence
$h^{\prime }\left(x\right)<0$ since $h^{\prime }\left(x\right)=f^{\prime }\left(g\right(x\left)\right).g^{\prime }\left(x\right)$ and $g^{\prime }\left(x\right)<0$.
Therefor $h\left(x\right)$ is a decreasing function too.
Hence $fog$ will be decreasing function.