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The domain of the function $f(x) = \sqrt{25-x^2}+\sqrt{x-2} + \frac{1}{\sqrt{1 - x}}$ is an interval of what width?

 Oct 7, 2023
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To find the domain of f(x)=25−x2​+x−2​+1−x​1​, we need to consider the following restrictions:

The first term, 25−x2​, is defined only when 25−x2≥0. This means that −5≤x≤5.

The second term, x−2​, is defined only when x−2≥0. This means that x≥2.

The third term, 1−x​1​, is defined only when 1−x>0. This means that x<1.

Therefore, the domain of f(x) is the intersection of the intervals [−5,5], [2,∞), and (−∞,1). The intersection of these intervals is the interval [2,5). This interval has a width of 5−2=3​.

Note: We can also find the domain of f(x) by graphing it. The graph shows that f(x) is defined only for values of x in the interval [2,5].

 Oct 7, 2023

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