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

Feb 24, 2017

The domain is $\left\{x | x > 1 , x \in \mathbb{R}\right\}$

#### Explanation:

The process here is to verify first the domain of $g \left(x\right)$ (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.

$f \left(g \left(x\right)\right) = \frac{1}{\sqrt{x - 1}}$

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

$\left\{x | x > 1 , x \in \mathbb{R}\right\}$

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

Hopefully this helps!