Show that is not onto (##\frac{x}{x^2+1}##)

  • Thread starter knowLittle
  • Start date
In summary: I hope this helps.In summary, we are trying to show that the function $$\frac{x}{x^2+1}$$ is either onto or not. The given domain is all real numbers except 0, and the range is all real numbers. To solve this, we can set up the equation y = $$\frac{x}{x^2+1}$$ and solve for x. By doing so, we get a quadratic equation which we can solve to determine whether there are any restrictions on y. If there are, then the function is not onto the entire real line. However, with the given domain, it becomes clear that the function is not surjective.
  • #1
knowLittle
312
3

Homework Statement


I need to show that $$\frac{x}{x^2+1}$$ is either onto or not.
My domain is $$R-{0}$$ and range is $$R$$

Homework Equations


I have learn to do this to show that a function is surjective
y = $$\frac{x}{x^2+1}$$ and solve for x, but I am not sure how to proceed here.

The Attempt at a Solution


By trial and error and a calculator I have seen that the range is $$\frac{1}{2} \leq x \leq \frac{1}{2}$$
It means that it's not onto.

Any help on how to proceed?
Thank you.
 
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  • #2
knowLittle said:

Homework Statement


I need to show that $$\frac{x}{x^2+1}$$ is either onto or not.
My domain is $$R-{0}$$ and range is $$R$$
That is not the domain, and the range is not all real numbers.
knowLittle said:

Homework Equations


I have learn to do this to show that a function is surjective
y = $$\frac{x}{x^2+1}$$ and solve for x, but I am not sure how to proceed here.
Solve the equation above for x (which you said). First thing to do is multiply both sides by x2 + 1.
knowLittle said:

The Attempt at a Solution


By trial and error and a calculator I have seen that the range is $$\frac{1}{2} \leq x \leq \frac{1}{2}$$
It means that it's not onto.

Any help on how to proceed?
Thank you.
 
  • #3
The question says that A= R-{0} and B =R. Then, that f:A ->B and I need to show whether they 1-1 and whether they are onto. Prove.
Thanks for the hint.
 
  • #4
Once, I have ##y(x^2+1) = x## what else can I do?
## y x^2 + y = x## divide all over x^2 wouldn't work.
 
  • #5
knowLittle said:
Once, I have ##y(x^2+1) = x## what else can I do?
## y x^2 + y = x## divide all over x^2 wouldn't work.
Move the x term over to the left side -- you have a quadratic in x, which you should know how to solve.

The reason for doing this (i.e., solving for x from the original equation) is to determine whether there are any restrictions on y. If there are, your function is not onto the entire real line (the y-axis). If there are no restrictions, the function is onto the reals. IOW, for any y value whatsoever, there is an x value that maps to it.
 
  • #6
knowLittle said:
The question says that A= R-{0} and B =R. Then, that f:A ->B and I need to show whether they 1-1 and whether they are onto. Prove.
For the function in this problem, there is no inherent reason for the domain to be R - {0}. The function is defined for all real numbers.
 
  • #7
Mark44 said:
For the function in this problem, there is no inherent reason for the domain to be R - {0}. The function is defined for all real numbers.
With the given domain, though, it becomes obvious that the function isn't surjective.
 
  • #8
Mark44 said:
For the function in this problem, there is no inherent reason for the domain to be R - {0}. The function is defined for all real numbers.
vela said:
With the given domain, though, it becomes obvious that the function isn't surjective.
It wasn't clear to me that that restriction was actually part of this problem.
 

FAQ: Show that is not onto (##\frac{x}{x^2+1}##)

What does it mean for a function to be onto?

A function is considered onto, or surjective, if every element in the range has at least one corresponding element in the domain. In other words, every output value has at least one input value that produces it.

How do you prove that a function is not onto?

To prove that a function is not onto, you must find at least one element in the range that does not have a corresponding element in the domain. In other words, there must be at least one output value that does not have an input value that produces it.

Can you provide an example of a function that is not onto?

Yes, the function ##f(x) = \frac{x}{x^2+1}## is an example of a function that is not onto. The range of this function is all real numbers except for 0, but there is no input value that produces an output of 0.

What is the graph of a function that is not onto?

The graph of a function that is not onto will have gaps or breaks in the line, indicating that there are output values that do not have corresponding input values. In the case of ##f(x) = \frac{x}{x^2+1}##, the graph will have a vertical asymptote at x=0, indicating that there is no output value of 0.

Why is it important to determine if a function is onto?

Determining if a function is onto is important because it helps us understand the relationship between the input and output values of a function. If a function is not onto, it means that there are some output values that cannot be produced, which may limit the usefulness of the function in certain situations.

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