Circle Equation

[harpazo]

If y = x^2 + 5x + 2, then we can say that y is written as a function of x. I understand that for each value of x, there is exactly one value of y.

Why is statement above NOT THE SAME for the equation y = x^2 + y^2? I know that y represents a circle here but the book is not too clear in terms of why this equation does not represent y written in terms of x. What do you say?

 

[pka]

If y = x^2 + 5x + 2, then we can say that y is written as a function of x. I understand that for each value of x, there is exactly one value of y. Why is statement above NOT THE SAME for the equation y = x^2 + y^2? I know that y represents a circle here but the book is not too clear in terms of why this equation does not represent y written in terms of x. What do you say?

In this case neither \(\displaystyle x^2\text{ nor }y^2\) is a function of the other. Do you see why?

 

[Dr.Peterson]

If y = x^2 + 5x + 2, then we can say that y is written as a function of x. I understand that for each value of x, there is exactly one value of y.

Why is statement above NOT THE SAME for the equation y = x^2 + y^2? I know that y represents a circle here but the book is not too clear in terms of why this equation does not represent y written in terms of x. What do you say?

When we say y is written in terms of x, we mean that the left-hand side is y alone, and the other side contains x as the only variable -- no y.

I would not say that y represents a circle; I'd say the entire equation represents a circle.

 

[harpazo]

In this case neither \(\displaystyle x^2\text{ nor }y^2\) is a function of the other. Do you see why?

This is why I am asking. Why is y = x^2 + y^2 not a function? Can it be that the circle fails the vertical line test?

 

[Deleted member 4993]

If y = x^2 + 5x + 2, then we can say that y is written as a function of x. I understand that for each value of x, there is exactly one value of y.

Why is statement above NOT THE SAME for the equation y = x^2 + y^2? I know that y represents a circle here but the book is not too clear in terms of why this equation does not represent y written in terms of x. What do you say?

That does not represent a circle! ...................INCORRECT

The equation for a circle, centered at the origin, is:

x² + y² = r²

Derived from above:

y = \(\displaystyle \sqrt{r^2 - x^2}\)

is a function (representing a semi-circle).

 

[harpazo]

That does not represent a circle!

The equation for a circle is:

x² + y² = r²

Derived from above:

y = \(\displaystyle \sqrt{r^2 - x^2}\)

is a function (representing a semi-circle).

Ok. No problem. Now, the question remains:

Why can't we express y as a function of x for y = x^2 + y^2?

 

[pka]

This is why I am asking. Why is y = x^2 + y^2 not a function? Can it be that the circle fails the vertical line test?

A function is a set of ordered pairs. That set has one most important one among others: no two pairs have the same first term.
In \(\displaystyle x^2+y^2=4\) both \(\displaystyle (1,\sqrt{3})~\&~(1,-\sqrt{3})\) are two pairs with the same first term. (That is the vertical line test.)

 

[Deleted member 4993]

Ok. No problem. Now, the question remains:

Why can't we express y as a function of x for y = x^2 + y^2?

Yes - the vertical line test.

However, derived from expression you can write:

y(x) = 1/2 + 1/2 * \(\displaystyle \sqrt{1- 4*x^2}\).............or..............

y(x) = 1/2 - 1/2 * \(\displaystyle \sqrt{1- 4*x^2}\)

Both of those are "functions".

 

[Deleted member 4993]

That does not represent a circle! ...................INCORRECT

The equation for a circle, centered at the origin, is:

x² + y² = r²

Derived from above:

y = \(\displaystyle \sqrt{r^2 - x^2}\)

is a function (representing a semi-circle).

y = x^2 + y^2

(y - 1/2)^2 + x^2 = (1/2)^2

The center of this circle is at (0,1/2) and radius = 1/2

 

[harpazo]

Thank you everyone.

 

[Otis]

… y = x^2 + y^2 … I know that y represents a circle here …

Symbol y is a variable, and it represents a number (not a circle).

Try to remember that.

 

[harpazo]

Symbol y is a variable, and it represents a number (not a circle).

Try to remember that.

What about f(x) = x^2 + y^2?

 

[Otis]

What about f(x) = x^2 + y^2?

That question is too vague to answer. Can you finish the question? (Please be complete. State specifically what you're trying to ask.)

\(\;\)

 

[harpazo]

That question is too vague to answer. Can you finish the question? (Please be complete. State specifically what you're trying to ask.)

\(\;\)

We cannot express y = x^2 + y^2 as a function of x. Why?

 

[pka]

We cannot express y = x^2 + y^2 as a function of x. Why?

To harpazo, your mathematics ignorance is on full display in that post.
Can you show us where one could ever, ever see \(\displaystyle y=x^2+y^2\)
Please show us a single instance anywhere that appears in print on the web or otherwise.
If you cannot then you will have a massive amount of egg on your face.

 

[harpazo]

To harpazo, your mathematics ignorance is on full display in that post.
Can you show us where one could ever, ever see \(\displaystyle y=x^2+y^2\)
Please show us a single instance anywhere that appears in print on the web or otherwise.
If you cannot then you will have a massive amount of egg on your face.

I made a huge typo in the original post. The equation should be 25 = x^2 + y^2. This is a circle for sure with radius 5. Here we cannot express y as a function of x.

 

[HallsofIvy]

What about f(x) = x^2 + y^2?

That notation makes no sense. "f(x)" says that f is a function depending on "x" alone but you have a "y" on the right side.

 

[Deleted member 4993]

To harpazo, your mathematics ignorance is on full display in that post.
Can you show us where one could ever, ever see \(\displaystyle y=x^2+y^2\)
Please show us a single instance anywhere that appears in print on the web or otherwise.
If you cannot then you will have a massive amount of egg on your face.

I think you are over-looking one situation (I did!):

y = x^2 + y^2

y^2 - y + x^2 = 0

(y - 1/2)^2 + x^2 = (1/2)^2 ............This is an equation of a circle of radius (1/2) and centered at (0,1/2)

 

[Dr.Peterson]

I made a huge typo in the original post. The equation should be 25 = x^2 + y^2. This is a circle for sure with radius 5. Here we cannot express y as a function of x.

You kept repeating the same 'typo", which doesn't help. And there are other wording issues that got in the way of a helpful discussion.

The important thing is, do you understand the answer to your question? Here is what you asked, as you apparently intended it, with three things changed:

If y = x^2 + 5x + 2, then we can say that y is written as a function of x. I understand that for each value of x, there is exactly one value of y.

Why is statement above NOT THE SAME for the equation 25 = x^2 + y^2? I know that the equation represents a circle here but the book is not too clear in terms of why this equation does not represent y as a function of x. What do you say?

Is that your question? What do you say now? And if you still have a question about it, can you quote what the book says, so we can clarify it for you?

 

[harpazo]

That notation makes no sense. "f(x)" says that f is a function depending on "x" alone but you have a "y" on the right side.

I said the original equation is a typo at my end. Here is the updated equation: 25 = x^2 + y^2. This is a circle with radius 5.