pre-calculus

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Determine whether y is a function of x:

x^2y-x^2+4y=0

• pre-calculus -

Do you mean "Can y be expressed as a function of x?". If so then the answer is yes it can, if by "x^2y" you mean "x²y" and not "x raised to the power of 2y". The equation x²y - x² + 4y = 0
can be written as (x²+4)y = x², from which you can easily express either x in terms of y, or y in terms of x.

• pre-calculus -

well, you can write y as a function of x, and get one and only one value of y for every x
however x is not a function of y because there are two values of x for every y.

• pre-calculus -

Fair enough, though you could presumably define a function by restricting the range to just zero plus either the positive or negative real numbers.

• pre-calculus -

• pre-calculus -

The answer is yes: you can write y as a function of x. Damon and I were debating whether you can write x as a function of y, which wasn't what you were asked.

• pre-calculus -

How exactly do you know that you can get one and only one value of y for every x?

• pre-calculus -

There's only one value for y because your original equation (x^2y-x^2+4y=0) can be rewritten as (x²+4)y = x². Or, by dividing both sides by (x²+4),

y = x²/(x²+4)

Feed any value of x into the right-hand side, and you'll get exactly one value for y.

• pre-calculus -

OH thanks, gotcha!