function help?

[eliT]

I have a question, and would appreciate help in answering it. Thank you.

"show that y^2 - xy - 2 = 0 cannot be a function ƒ(x) for x ∈ R."

 

[JeffM]

I have a question, and would appreciate help in answering it. Thank you.

"show that y^2 - xy - 2 = 0 cannot be a function ƒ(x) for x ∈ R."

That is a very strange question. Is it complete? Is there some context to it?

Consider the situation if y = 0. What value of x can possibly satisfy the equation \(\displaystyle 0^2 - x * 0 - 2 = 0?\)

 

[eliT]

It's the entire question

I know that I can show that for, say, a value of 3 or -10 that the discriminant will be greater than zero (and thus there will be more than one value for y) but I have absolutely no idea how to demonstrate that it is true for all values of x.

 

[DrPhil]

It's the entire question

I know that I can show that for, say, a value of 3 or -10 that the discriminant will be greater than zero (and thus there will be more than one value for y) but I have absolutely no idea how to demonstrate that it is true for all values of x.

If there is ANY value of x ∈ R for which you can't express y as a function of x, then the question is satisfied.

 

[eliT]

I've spent hours trying to understand. I've done tons of questions (correctly) but I just don't know what to write as the answer to this question. I was seriously hoping someone could tell me what equation or statement actually answers the question that was asked. But so far on three different forums the only replies I get are people asking me questions, giving vague hints I don't understand or telling me to research functions. I'm fairly close to tears now... so tired of trying and only failing to understand.

 

[lookagain]

I have a question, and would appreciate help in answering it. Thank you.

"show that y^2 - xy - 2 = 0 cannot be a function ƒ(x) for x ∈ R."

You can solve for y in terms of x using the Quadratic Formula.


\(\displaystyle y^2 - xy - 2 = 0\)


\(\displaystyle a = 1, \ \ b = -x, \ \ c = -2\)



\(\displaystyle y \ = \ \dfrac{ \ -(-x) \pm \sqrt{(-x)^2 \ - \ 4(1)(-2) \ }}{2(1)} \ \implies\)




\(\displaystyle y \ = \ \dfrac{ \ x \pm \sqrt{x^2 \ + 8 \ }}{2} \)



With the plus or minus feature, and the fact that the radicand can never equal 0, this expression necessarily
assigns two distinct values of y to every x.

But it only needs to occur one place where two distinct y-values are assigned to one x-value for it not to be a function.

 

[DrPhil]

I've spent hours trying to understand. I've done tons of questions (correctly) but I just don't know what to write as the answer to this question. I was seriously hoping someone could tell me what equation or statement actually answers the question that was asked. But so far on three different forums the only replies I get are people asking me questions, giving vague hints I don't understand or telling me to research functions. I'm fairly close to tears now... so tired of trying and only failing to understand.

You have done all you need to do. You have found real values of x for which y is double-valued, hence NOT a function of x. Thus you have shown that y is not a function on the domain of all real numbers.