If function $f$ be strictly decreasing and function $g$ be strictly increasing on $R_f$ which statement is true?
$1) g\circ f\quad \text{is a constant function}$
$2) g\circ f\quad \text{is neither increasing nor decreasing}$
$3) g\circ f\quad \text{is strictly increasing}$
$4) g\circ f\quad \text{is strictly decreasing}$
I think $g\circ f$ is strictly increasing because the input of this function is $R_f$ (range of $f$) and $g$ is strictly increasing on $R_f$ as stated in the questino. But I doubt this answer is correct because I didn't use the first part of the question which says $f$ is strictly decreasing function.
Best Answer
If $x>y$, then $f(x)<f(y)$ because $f$ is strictly decreasing. Let’s write $a=f(x)$ and $b=f(y)$. We have $a<b$, so $g(a)<g(b)$ because $g$ is strictly increasing. Hence, if $x>y$, then $(g \circ f)(x)<(g \circ f)(y)$.
$g \circ f$ is strictly decreasing.