Question

Let $f$ be an increasing function on $[a,b]$ and $g$ be a decreasing function on $[a,b]$, then on $[a,b]$.

• A. $fog$ is a decreasing function.
• B. $gof$ is an increasing function.
• C. $fog$ is an increasing function.
• D. None of these

Answer 1

Answer: (A), (B)

The correct options are
A $fog$ is a decreasing function.
B $gof$ is an increasing function.

Let $x_1$,$x_2$ $\in$ [a,b],
Since g is decreasing
For $x_1<x_2$,
$\Rightarrow g\left(x_1\right)>g\left(x_2\right)$
$\Rightarrow f\left(g\right(x_1\left)\right)>f\left(g\right(x_2\left)\right)$
i.e. fog is decreasing function on [a,b] .
Since f is increasing
For $x_1<x_2$,
$\Rightarrow f\left(x_1\right)<f\left(x_2\right)$
$\Rightarrow g\left(f\right(x_1\left)\right)<g\left(f\right(x_2\left)\right)$
i.e. gof is increasing function on [a,b] .