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Oksanka [162]
3 years ago
11

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 "c

over" information to break your proof(s) into cases. part of (0, oo).. you may want to use this
Mathematics
1 answer:
solniwko [45]3 years ago
3 0

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.

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