One-to-one function determination

[ehrenfest]

Homework Statement

Without a graphing calculator, how can you tell that the function

f(x) = x/(x^2+1) is one-to-one?

Homework Equations

The Attempt at a Solution

You can sketch both x and 1/(x^2+1) separately but I did not think it was obvious that when you multiplied them togethor the result was not one-to-one.

 

[Hurkyl]

Is your homework problem actually to show that that function is one-to-one? One way to show that a function is one-to-one is to start by stating the definition of one-to-one, and then prove that this function satisfies the definition.

 

[ehrenfest]

A function is one-to-one if whenever s1 and s2 are two different elements in the domain, f(s1) is not equal to s2.

 

[Hurkyl]

The contrapositive is often easier to work with; if f(x)=f(y), then x=y.

 

[Emad]

It is one-to-one function so that assuming that
f(a)=f(b)
we find easily that a=b.
you can follow the link to see the graph

http://www.4shared.com/file/d8TFg6zC/emad.html

 

[ehild]

Find what value/values of x belong to a certain value of y. ehild

 

[hunt_mat]

You are asked to show:
$$\frac{a}{1+a^{2}}=\frac{b}{1+b^{2}}$$
So expand and write as a quadratic:
$$b^{2}-\Bigg( a+\frac{1}{a}\Bigg) b+1=0$$
If f(x) is one to one, the above quadratic should have one and only one solution, does it?

 

[vela]

If you flip the function over, you get

$$y=\frac{x}{1+x^2} \Rightarrow \frac{1}{y} = x+\frac{1}{x}$$

That might be a bit easier to analyze.

 

[hunt_mat]

You can factorise, my quadratic equation.

Mat

 

[vela]

Yeah, I know. I was just offering yet another way to look at the problem. I hadn't really thought about the problem until I saw the linear term in your quadratic and realized you could easily deduce the answer looking at the reciprocal of the function.

 

[hunt_mat]

I know, I was only winding you up.

 

[awkward]

What is f(0)?

What is $$\lim_{x \to \infty} f(x)$$?

What does this tell you?

 

[hunt_mat]

Or even(coming from my quadratic equation) what is f(a) and f(1/a)?