If f and g are two increasing functions such that $gof$ is defined, then

Let $x_1,x_2\in R$ such that $x_1<x_2$ . Then,

$\begin{array}{l}{x}_1<x_2\\ \Rightarrow f\left(x_1\right)<f\left(x_2\right) [·: f is an increa\sin g function]\\ \Rightarrow g\left(f\left(x_1\right)\right)<g\left(f\left(x_2\right)\right) [·: g is an increa\sin g function]\\ \Rightarrow g\circ f\left(x_1\right)<g\circ f\left(x_2\right)\end{array}$

Hence, $gof$ is an increasing function.