What is the continuity of the composite function #f(g(x))# given #f(x)=1/sqrtx# and #g(x)=x-1#?

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Answer 1 · 2017-02-24T14:19:42

The domain is #{x| x > 1, x in RR}#

The process here is to verify first the domain of #g(x)# (the inner function in the composition).

This is a linear function, so is defined on all values of #x# within its domain. We now consider the composition.

#f(g(x)) = 1/sqrt(x - 1)#

There will be two types of restriction on the domain in this problem.

•When the number underneath the #√# is less than #0#.
•When the denominator equals #0#.

The number under the square root will be negative whenever #x < 1#. The denominator will equal #0# when #x = 1#, so the domain of the composition is

#{x| x > 1, x in RR}#

In other words, the composition is continuous for all values of #x# that are larger than #1#.

Hopefully this helps!