Determine whether y is a function of x:
x^2y-x^2+4y=0
Please help. Thanks.
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.
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.
Fair enough, though you could presumably define a function by restricting the range to just zero plus either the positive or negative real numbers.
So the answer is yes?
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.
How exactly do you know that you can get one and only one value of y for every x?
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.
OH thanks, gotcha!
To determine whether y is a function of x, we need to check if there is a unique y-value for each x-value, or if there are multiple y-values for the same x-value.
To do this, we can rearrange the given equation into the form y = f(x).
Given equation: x^2y - x^2 + 4y = 0
Rearranging, we can gather the terms involving y on one side:
x^2y + 4y = x^2
Factoring out y on the left side:
y(x^2 + 4) = x^2
Now we can solve for y by dividing both sides by (x^2 + 4):
y = x^2 / (x^2 + 4)
This equation represents y as a function of x, as it gives a unique y-value for each x-value. Hence, y is a function of x.