Question

a) Prove that a strictly decreasing function from R to itself is one-to-one.

b) Give an example of a decreasing function from R to itself is not one-to-one.

Short answer

$f\left(x\right)=0\text{for}x<0\text{and}f\left(x\right)=x\text{for}x\ge 0$

Step-by-step solution

Step: 1

Let f be a strictly decreasing function from R to itself.

If a < b, then f(a) > f(b) if a > b then f(a) < f(b)

Thus if $a\ne b$, then$f\left(a\right)\ne f\left(b\right)$

Step: 2

Therefore,

$f\left(x\right)=0\text{for}x<0\text{and}f\left(x\right)=x\text{for}x\ge 0$