# Thread: z As Function of Two Variables

1. ## z As Function of Two Variables

Determine if z is a function of x and y.

1. xz^2 + 2xy - y^2 = 4

I say yes because there is only one z letter in the equation.

2. z + x lny - 8 = 0

I have no idea about number 2.

2. ## Re: z As Function of Two Variables

Originally Posted by USNAVY
Determine if z is a function of x and y.

1. xz^2 + 2xy - y^2 = 4

I say yes because there is only one z letter in the equation.
So what? What is your definition of "function"?

2. z + x lny - 8 = 0

I have no idea about number 2.
z= 8- x ln(y)

3. ## Re: z As Function of Two Variables

Originally Posted by HallsofIvy
So what? What is your definition of "function"?

z= 8- x ln(y)
I will get back to you on this post when time allows.

4. ## Re: z As Function of Two Variables

Originally Posted by USNAVY
Determine if z is a function of x and y.

1. xz^2 + 2xy - y^2 = 4

I say yes because there is only one z letter in the equation.
Echoing HallsofIvy's response, how are you defining a function? In the equation, for example, $z^2 = y$ there is "only one z in the equation" but my example is not a function.

-Dan

5. ## Re: z As Function of Two Variables

Originally Posted by topsquark
Echoing HallsofIvy's response, how are you defining a function? In the equation, for example, $z^2 = y$ there is "only one z in the equation" but my example is not a function.

-Dan
I will reply in full tomorrow.

6. ## Re: z As Function of Two Variables

Do you know the vertical line test for a function?

7. ## Re: z As Function of Two Variables

Originally Posted by chiro
Do you know the vertical line test for a function?
For a 3D problem like this, you would have to think about using a horizontal plane test...

8. ## Re: z As Function of Two Variables

Originally Posted by chiro
Do you know the vertical line test for a function?
Yes, of course.

9. ## Re: z As Function of Two Variables

That is correct for higher dimensions.

Also - the implicit derivatives should be separable if you can have a function.

10. ## Re: z As Function of Two Variables

Originally Posted by HallsofIvy
So what? What is your definition of "function"?

z= 8- x ln(y)
A function relates an input to an output.

11. ## Re: z As Function of Two Variables

You still have a vertical line test for a function but you will do it in the z-axis not the x or y.

12. ## Re: z As Function of Two Variables

Originally Posted by USNAVY
A function relates an input to an output.
A function relates an input to a unique output.

13. ## Re: z As Function of Two Variables

I will post my answer to question 1 later.

14. ## Re: z As Function of Two Variables

I will post my answer to question 2 later.