What is the domain of f(x)?

Question

#1/(sqrt(x-7)*(x^2-4)#

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Answer 1 · 2017-12-31T20:36:34

#x > 7 : x \in \RR#

You have to avoid #x# values that makes the denominator zero.
So:
#x^2-4 \ne 0#
#sqrt(x-7) > 0#

Let's solve the first one:
#x^2-4 = 0#
#x^2=4#
#x\ne\pm2#

While the second one:
#sqrt(x-7) > 0#
#x-7>0#
#x>7#

The final results are #x>7# and #x\ne \pm2#, but the first is stronger so the domain is:
#x > 7 : x \in \RR#

graph{1/(sqrt(x-7)*(x^2-4)) [4.32, 10.482, -0.642, 2.436]}