Circle Equation

[harpazo]

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:

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?

Dr. Peterson,

Here is what the book makes known:

If y = x^2 + 5x + 2, then y is a function of x. For each value of x there is one and only one value of y. However, if the equation x^2 + y^2 = 25 defines the correspondence between x and y, then y is not a function of x.

My Question:

Based on what is stated above, why is y not a function of x for the equation x^2 + y^2 = 25?

 

[Dr.Peterson]

I think the question has been answered (though with reference to a different equation), which is why I asked if you thought so, too.

The equation [MATH]x^2 + y^2 = 25[/MATH] does not represent y as a function of x because for a given value of x, there can be more than one value of y. That can't be true for a function, according to its definition; a function must have a single output for a given input.

One way to see that is to know that it is the equation of a circle, which fails the vertical line test.

Another is to try solving for y; you end up with [MATH]y = \pm\sqrt{25 - x^2}[/MATH], which has two values.

Another is just to see that whatever y is, if you replace it with its opposite, the equation will still be true, because [MATH]y^2 = (-y)^2[/MATH].

 

[Otis]

Circles on the xy-plane do not represent a functional relationship between x and y because two different values of y satisfy the equation for nearly every value of x along the horizontal diameter. (The point at each end are the only exceptions.)

Let's pick a value for x, to demonstrate. This circle's center is at the Origin, and its radius is 5 units. Therefore, the values of x along the horizontal diameter run from -5 to +5.

Let x = 4
Let y = -3

25 = (4)^2 + (-3)^2
25 = 16 + 9
25 = 25

Now let y = 3

25 = (4)^2 + (3)^2
25 = 16 + 9
25 = 25

That shows one value of x works with two different values of y, to satisfy the equation. But a functional relationship allows only one value of y for each x. Therefore, the given equation does not represent a function.

If we want to describe a circle on the xy-plane using functions, then we need two different functions. We divide the circle in half, along the horizontal diameter. This way, the upper half of the circle is a function, and the lower half is also a function -- but the two functions are not the same function (each of them produce a different half of the circle).

As shown previously, we solve the given equation for y, to produce the two functions:

f(x) = sqrt(25 - x^2)

g(x) = -sqrt(25 - x^2)

Note that functions f and g are the same, except for the sign on the output. Can you tell, just by looking, which function corresponds to the lower half of the circle?

?

 

[harpazo]

I think the question has been answered (though with reference to a different equation), which is why I asked if you thought so, too.

The equation [MATH]x^2 + y^2 = 25[/MATH] does not represent y as a function of x because for a given value of x, there can be more than one value of y. That can't be true for a function, according to its definition; a function must have a single output for a given input.

One way to see that is to know that it is the equation of a circle, which fails the vertical line test.

Another is to try solving for y; you end up with [MATH]y = \pm\sqrt{25 - x^2}[/MATH], which has two values.

Another is just to see that whatever y is, if you replace it with its opposite, the equation will still be true, because [MATH]y^2 = (-y)^2[/MATH].

You nailed it. Thank you.

 

[harpazo]

Circles on the xy-plane do not represent a functional relationship between x and y because two different values of y satisfy the equation for nearly every value of x along the horizontal diameter. (The points at each end are the only exceptions.)

Let's pick a value for x, to demonstrate. This circle's center is at the Origin, and its radius is 5 units. Therefore, the values of x along the horizontal diameter run from -5 to +5.

Let x = 4
Let y = -3

25 = (4)^2 + (-3)^2
25 = 16 + 9
25 = 25

Now let y = 3

25 = (4)^2 + (3)^2
25 = 16 + 9
25 = 25

In the given equation's relationship between x and y, one value of x allows two different values of y to satisfy the equation. But in a functional relationship between x and y, the function allows only one value of y for each x. Therefore, the given equation does not represent a function.

If we want to describe a circle on the xy-plane using functions, then we need two different functions. We divide the circle in half, along the horizontal diameter. This way, the upper half of the circle is a function, and the lower half is also a function -- but the two functions are not the same function (each of them produce a different half of the circle).

As shown previously, we solve the given equation for y, to produce the two functions:

f(x) = sqrt(25 - x^2)

g(x) = -sqrt(25 - x^2)

Note that functions f and g are the same, except for the sign on the output. Can you tell, just by looking, which function corresponds to the lower half of the circle?

?

Great notes. Thank you.