Determine whether z is a function of x and y?

Question

1 (x^2)z + yz - xy = 10

2. (x^2)/4 + (y^2)/9 + z^2 = 1

Just have to determine and prove whether or not z is a function of x and y. I'm kind of stumped as there are no examples I could find anywhere.

Forgot to mention my book said 1. z is a function of x and y and 2. z IS NOT a function of x and y.

I solved each for z but was really confused with the second one.

Answer 1

You can think of a function as something that gives you at most one output value for a given input value. So in these cases, you can think of a function as something that gives you at most one value for z given an x and a y. So given that:

1. Yes. You can factor out z and rearrange terms as follows:

z(x^2 + y) - xy = 10
z(x^2 + y) = 10 + xy
z = (10 + xy) / (x^2 + y)

So since we were able to solve for an expression involving x and y that gives us one unique z, z is a function of x and y.

[ADDENDUM: demiurge42 brings up a good point, that there is a value of (x, y) for which z could be anything. So you might want to check that with your teacher to see if the book is wrong--or if that case was specifically mentioned in the problem.]

2. No. If you try following the same steps, you get

z^2 = 1 - (x^2)/4 + (y^2)/9
z = +/- sqrt(1 - (x^2)/4 + (y^2)/9)

(Remember that if you solve for z but you have z^2, then the answer is +/- sqrt(), and not just sqrt(), since z doesn't have to be positive.)

But we don't always get a unique value for z given an x and a y--when (x^2)/4 + (y^2)/9 = 1, z is zero, but for all other values of x and y, z can be positive or negative. So here z is not a function of x and y.

Answer 2

1. z is NOT a function of x and y.

Factor the left side:
z(x^2 + y) -xy = 10
Let x = 10^(1/3) and y = -10^(2/3).
then x^2 + y = 0 and -xy = 10.
It doesn't matter what z is since it is multiplied by zero.
Since z has more than one possible value for some (x,y), it isn't a function of x and y.

Dividing both sides by (x^2 + y) removes some valid values for x and y. Namely values when x^2 + y = 0.

2. Also not a function. Since you get z = ± √(1-x^2/4-y^2/9)

Answer 3

If you write z (x,y) then z is a function of x and y.

1. z ( x^2 + y) = 10+xy
z = (10 + xy) / (x^2 + y)


IT IS A FUNCTION OF x and y.

2. z^2 = 1 - (x^2)/4 - (y^2)/9
z = sqrt(1 - (x^2)/4 - (y^2)/9)

IT IS A FUNCTION OF x and y too.

The first example seems like a paraboloid,.
The second part seems like an ellipsoid.

Answer 4

solve for z

z(x^2 + y) = 10 + xy

z = (10 + xy)/ (x^2 + y)

XXXXXXXXXXXXXXX

(x^2)/4 + (y^2)/9 + z^2 = 1

z^2 = 1-((x^2)/4 + (y^2)/9 )

z = +/1 square root of [1-(x^2)/4 - (y^2)/9)]


x^2 and y^2 will always be positive so they will not give a unique value for z

Answer 5

Omega