Question

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

Answer 1

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

Explanation:

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!