f is strictly increasing and g is strictly decreasing on R. Which describes h(x) = f(g(x))?

Question 9

MediumOriginal question

$f$ is strictly increasing and $g$ is strictly decreasing on $\mathbb{R}$ . Which describes $h(x) = f(g(x))$ ?

Strictly increasingCorrect answerYour answerStrictly decreasingCorrect answerYour answerConstantCorrect answerYour answerIt depends on

f

and

g

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

Take $x_1 < x_2$ . As $g$ is strictly decreasing, $g(x_1) > g(x_2)$ .
As $f$ is strictly increasing, it keeps that order: $f(g(x_1)) > f(g(x_2))$ . So $h$ is strictly decreasing.

Worth noting

One decreasing step reverses the order once, and the increasing step keeps it, so the order ends reversed: try $f(x) = x$ , $g(x) = -x$ .

Ten questions from Increasing & Decreasing Functions, at your own pace.