Is f(x) = (x^2-2x+2)/(2x^2+2x+2) Surjective?

  • Thread starter Dollydaggerxo
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In summary, the conversation discusses determining the injectivity and surjectivity of a given function, with the focus on surjectivity. The individual has concluded that the function is neither injective nor surjective through various methods such as graphing and considering the range of values. The conversation also mentions completing the squares to make the analysis neater. Ultimately, it is determined that the function is neither injective nor surjective as it cannot be mapped onto every real number.
  • #1
Dollydaggerxo
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Find whether the following function is injective and/or surjective:
f(x)= (x^2-2x+2)/(2x^2+2x+2)

Basically, I just need to know if it is surjective. I don't want any proof because I should work it out myself seen as it is homework haha. I have a done it a few different ways and came out with the answer that it is neither, however, the way the question is worded seems like it should be one or the other or both.
Any help would be greatly appreciated.
Thank you
 
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  • #2
Welcome to PF!

Hi Dollydaggerxo! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)
Dollydaggerxo said:
… I have a done it a few different ways and came out with the answer that it is neither, however, the way the question is worded seems like it should be one or the other or both.

Show us how you got your answer, and then we'll see what went wrong. :wink:
 
  • #3
oh i didnt see that haha thanks :)

and well, I constructed the graph of it, and since there is a horizontal line that intersects it more than once, I have concluded that it is not injective.

for the surjectvity, I have it isn't surjective because if you were to construct a horizontal line at say, -1, it wouldn't intersect it. Therefore it is not surjective for R -> R but perhaps for R -> R+o
also for the surjectivity, I found that there is no real number, x, such that the function = -1
(i apologise for the rubbish symbols!)
 
  • #4
ok but messy …

in particular, for the surjectivity, it would be neater to complete the squares, and then consider whether certain things can be negative. :wink:
 
  • #5
oh yes i could do that...
neither could be negative, which means its not surjective because a positive/positive cannot be negative and therefore cannot be mapped onto every real number?

sorry for all the questions, I appreciate your help.

so is it neither? was that right or not?
Thank you
 
  • #6
Dollydaggerxo said:
neither could be negative, which means its not surjective because a positive/positive cannot be negative and therefore cannot be mapped onto every real number?

Yup! :biggrin:
so is it neither? was that right or not?

Yes, that was right. :smile:
 
  • #7
Okay, thanks for your help, greatly appreciated! :)
 

FAQ: Is f(x) = (x^2-2x+2)/(2x^2+2x+2) Surjective?

What is injectivity?

Injectivity is a property of a function in mathematics where each element in the domain maps to a unique element in the range. In other words, no two elements in the domain can map to the same element in the range. This is also known as the one-to-one correspondence.

How is injectivity determined?

To determine if a function is injective, you can use the horizontal line test. This involves drawing horizontal lines across the graph of the function. If a horizontal line intersects the graph at more than one point, then the function is not injective. If every horizontal line intersects the graph at most once, then the function is injective.

What is surjectivity?

Surjectivity is a property of a function in mathematics where every element in the range has at least one corresponding element in the domain. In other words, the function covers or maps onto the entire range.

How is surjectivity determined?

To determine if a function is surjective, you can use the vertical line test. This involves drawing vertical lines across the graph of the function. If a vertical line intersects the graph at more than one point, then the function is not surjective. If every vertical line intersects the graph at least once, then the function is surjective.

Can a function be both injective and surjective?

Yes, a function can be both injective and surjective. This type of function is known as a bijection. It means that every element in the domain maps to a unique element in the range and every element in the range has at least one corresponding element in the domain. In other words, the function has a one-to-one correspondence between the domain and range.

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