Question

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

Answer:

$f\left(x\right)=0$for$x<0$ androle="math" localid="1668404903804" $f\left(x\right)=-x$ for$x⩾0$

Step-by-step solution

Step: 1

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

If $a<b$, then $f\left(a\right)>f\left(b\right)$if $a>b$then$f\left(a\right)<f\left(b\right)$

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

Step: 2

Therefore,

$f\left(x\right)=0$for $x<0$and$f\left(x\right)=-x$ for$x⩾0$