Question

f(x)=(x+3)/(4-√(x^2-9))
what is the domain

Answer 1

you don't want the bottom to be zero
you also don't want the thingy under the square root to be negative
so first when is the bottom 0
\[4-\sqrt{x^2-9}=0 => 4=\sqrt{x^2-9} =>16=x^2-9\]
\[=>x^2-25=0 => x=\pm 5\]
so fraction is undefined when x=5 or x=-5
now what is the domain for sqrt{x^2-9}
so we have
that we need x^2-9 to be positive or equal to zero
x^2-9 =0 when x=-3 or x=3

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-3 3
test intervals around these numbers
+ - +
so the domain of sqrt{x^2-9} is (-inf,-3] U [3,inf)
but we do not include -5 or 5 for f since that would make f undefined
so we say (-inf,-5)U (-5,-3] U[3,5) U (5,inf)

Answer 2

thanks!! i appreciate your help <3