How do you determine the domain and range of the relation x^2 + y^2 = 1 and x^2 + y^2 = 100? i know it makes a circle but I'm just confused about the steps that you have to take in order to find the domain and range.  thankss in advance!

Expert Answers

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You should notice that each given equation describes a circle centered at origin, with the radius r.

You need to remember the equation of the circle centered at origin such that:

`x^2 + y^2 = r^2`

Comparing the first equation, `x^2 + y^2 =1` , to the general equation of the circle,`x^2 + y^2 = r^2` , yields that `r^2 = 1 => r = 1` .

Since the circle is centered at origin, its diameter is of length 2, meaning that the domain and the range are the same, `[-1,1].`

Comparing the first equation, `x^2 + y^2 = 100` , to the general equation of the circle, `x^2 + y^2 = r^2`  , yields that`r^2 = 100 ` `=> r = 10` .

Since the circle is centered at origin, its diameter is of length 20, meaning that the domain and the range are the same, `[-10,10].`

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For x^2+y^2=1, you already know the graph is a circle centered at (0,0) with radius 1. Think of squeezing the graph down onto the x axis. It'll overlap the axis when -1<=x<=1, or [-1,1] in interval notation, so that's the domain.

Do something similar for the range, except now compress the graph onto the y axis. It overlaps from -1<=y<=1, so that's the range.

The same idea applies to x^2+y^2=100. This is a circle centered at the origin with radius 10, so the domain and range will now both be [-10,10].

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