Question

Which of the following functions are monotonically decreasing functions ?

• A. $f\left(x\right)=2x+3$
• B. $f\left(x\right)=5-8x$
• C. $f\left(x\right)=3$
• D. None of these

Answer 1

Answer: (B), (C)

$f\left(x\right)=3$

If f(x) is monotonically decreasing, then for every $x_1,x_2$ $\in$ D where D is the domain of f(x),

$x_1>x_2\Rightarrow f\left(x_1\right)\le f\left(x_2\right).$

We will go through each options and check if it satisfied

A) $x_1>x_2\Rightarrow f\left(x_1\right)\le f\left(x_2\right)$

Or $x_1>x_2$ $\Rightarrow$ $2x_1+3\le 2x_2+3$

$\Rightarrow$ $2x_1\le 2x_2$

$\Rightarrow$ $x_1\le x_2$

Which is a contradiction. So the given function is not correct

B) $x_1>x_2$ $\Rightarrow$ $5-8x_1\le 5-8x_2$

$\Rightarrow$ $x_1\le x_2$, which is true. This is an example of monotonically decreasing function

C) $x_1>x_2$ $\Rightarrow$ $3\le 3$, which is always true. So C is also correct

Options B and C are correct