How would you find the domain and range of a circle on a graph whose points on the y axis are 5 and -5, and whose x axis coordinates are 8 and -8?

The curve you describe is not a circle, it could be an ellipse. Here is the curve I think you meant:

graph{x^2/64+y^2/25=1 [-20.27, 20.28, -10.14, 10.13]}

The domain is the set of all numbers for which there is a point on the curve with that #x#-value.
The range is the set of all numbers for which there is a point on the curve with that #y#-value.

It might be helpful to imagine squashing the graph down onto the #x#-axis to find the domain. The squashing it onto the #y#-axis to find the range.

For this graph, there are clearly no points with #x=-20# or #x=-10#,
in fact the least number that appears as a, #x#-coordnate of a point on this graph is #-8#. The greates is #8# and there is a point on the curve for every number between #-8# and #8#. Therefore the Domain is #[-8, 8]#

By similar reasoning, the range is #[-5, 5]#

(Be careful to read the #y#-values from least to greatest -- from bottom to top.)

Here's another example:

Find the domain and range of the equation whose graph is:

graph{x^2/100+y^2/4=1 [-14.24, 14.25, -6.21, 8.03]}

I hope you got Domain = #[-10, 10]# and
Range = #[-2, 2]#

One more example:

Find the domain and range of the equation whose graph is below.
Remember that the domain is all the #x# values used and the range is the #y#-values used. You can zoom in if you use your mouse wheel.

graph{(x+3)^2/25+(y-2)^2/4=1 [-12.515, 9.995, -4.18, 7.07]}

It looks like we use all the #x#-values from #-8# to #2#

So the domain is #[-8, 2]#

Now what about the range?

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I hope you got Range is #[0, 4]#, because that is correct.