Question

Let f and g be decreasingfunctions for all real numbers. Prove
that f o g isincreasing.

Answer 1

Answer with Step-by-step explanation:

We are given that two functions f and g are decreasing function for all real numbers .

We have to prove that fog is increasing.

Increasing Function: If
`x_1 \leq x_2`

Then,
`f(x_1)\leq f(x_2)`

Decreasing function: if
`x_1\leq x_2`

Then,
`f(x_1)\geq f(x_2)`

Suppose
`f(x)=-x`

`g(x)=-2x`

fog(x)=f(g(x))=f(-2x)=-(-2x)=2x

Therefore, f and g are decreasing function for all real numbers then fog is increasing.

Hence, proved