One-to-one function determination

AI Thread Summary
To determine if the function f(x) = x/(x^2+1) is one-to-one, one can apply the definition of a one-to-one function, which states that if f(a) = f(b), then a must equal b. A quadratic equation derived from setting f(a) equal to f(b) can be analyzed to check for unique solutions, indicating whether the function is one-to-one. Additionally, examining the behavior of the function as x approaches infinity and considering the reciprocal can provide further insights. The discussions suggest that analyzing these aspects can help confirm the function's one-to-one nature. Ultimately, the function is one-to-one as demonstrated through these mathematical approaches.
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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.
 
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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.
 
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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.
 
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The contrapositive is often easier to work with; if f(x)=f(y), then x=y.
 
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
 
Find what value/values of x belong to a certain value of y. ehild
 
You are asked to show:
<br /> \frac{a}{1+a^{2}}=\frac{b}{1+b^{2}}<br />
So expand and write as a quadratic:
<br /> b^{2}-\Bigg( a+\frac{1}{a}\Bigg) b+1=0<br />
If f(x) is one to one, the above quadratic should have one and only one solution, does it?
 
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.
 
You can factorise, my quadratic equation.

Mat
 
  • #10
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.
 
  • #11
I know, I was only winding you up.
 
  • #12
What is f(0)?

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

What does this tell you?
 
  • #13
Or even(coming from my quadratic equation) what is f(a) and f(1/a)?
 
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