Proof 8.3,4,5

Author:Kayla Moore

3. Prove that f is an onto function or prove that it is not.

Proof: Define

where

. Pick a point

. In order to be considered onto, there should be a point

for which

This means

and

. By solving for

and

, we know that

and

. Notice,

. Therefore, we have found the preimage of

. Since there is a point in the preimage of

that maps to an arbitrary point, we can conclude that the function

is onto.

Proof: Define

where

. Pick a point

. In order to be considered onto, there should be a point

for which

. This means

and

. By solving for

and

, we know that

and

. Notice,

. Therefore, we have found the preimage of

. Since there is a point in the preimage of

that maps to an arbitrary point, we can conclude that the function

is onto.

4. Prove that f is a one-to-one function or prove that it is not.

Proof: Define

. Pick two points

and

. If

then it is implied that

. If we consider the x and y coordinates separately, we notice that

and

which implies

from the previous statement. From this we see,

. Since

and

, we know

is one-to-one.

Proof: Define

. Pick two points

and

. Notice if

, then it is implied that

. If we consider the x and y coordinates separately, we notice that

which implies

and

which implies

. Since

and

, we know that

is one-to-one.

5. Prove or disprove that f is distance-preserving.

Proof: Define

. Consider points

and

. Determine the distance between the points using the distance formula. Note, the distance from

. The distance between

and

. If the function is in fact distance-preserving, these distances would be equivalent. Notice,

Since the sides are not equal to one another, we know that the function is not distance-preserving.

Proof: Define

. Consider the points

and

. Determine the distance between the points using the distance formula. Note, the distance from

. The distance between

and

. If the function is distance-preserving, then the distances would be the same. Notice,

. Since the distances are equivalent, we know that the function is distance preserving.

Proof 8.3,4,5

3. Prove that f is an onto function or prove that it is not.

4. Prove that f is a one-to-one function or prove that it is not.

5. Prove or disprove that f is distance-preserving.

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