Question

15 pts. Prove that the function f from R to (0, oo) is bijective if - f(x)=x2 if r- Hint: each piece of the function helps to "cover" information to break your proof(s) into cases. part of (0, oo).. you may want to use this

Answer 1

Answer with explanation:

Given the function f from R  to $(0,\infty)$

f: $R\rightarrow(0,\infty)$

$-f(x)=x^2$

To prove that  the function is objective from R to  $(0,\infty)$

Proof:

$f:(0,\infty )\rightarrow(0,\infty)$

When we prove the function is bijective then we proves that function is one-one and onto.

First we prove that function is one-one

Let $f(x_1)=f(x_2)$

$(x_1)^2=(x_2)^2$

Cancel power on both side then we get

$x_1=x_2$

Hence, the function is one-one on domain [tex[(0,\infty)[/tex].

Now , we prove that function is onto function.

Let - f(x)=y

Then we get $y=x^2$

$x=\sqrt y$

The value of y is taken from $(0,\infty)$

Therefore, we can find pre image  for every value of y.

Hence, the function is onto function on domain $(0,\infty)$

Therefore, the given $f:R\rightarrow(0.\infty)$ is bijective function on $(0,\infty)$ not on  whole domain  R .

Hence, proved.