Question

Discuss the continuity of the composite function \(h(x)=f(g(x))\). $$ \begin{array}{l} f(x)=\frac{1}{\sqrt{x}} \\ g(x)=x-1 \end{array} $$

Short answer

The function h(x) = \( f(g(x)) = \frac{1}{\sqrt{x-1}} \) is continuous for all x in the interval \( (1, +\infty) \).

Step-by-step solution

Identify Function Domains

Firstly, it is important to identify the domains of the given function. The function f(x) = \( \frac{1}{\sqrt{x}} \) is defined only for \( x > 0 \). The function g(x) = x-1 is defined for all real numbers.

Form Composite Function

Next, form the function h(x) = f(g(x)). Substitute the expression for g(x) into f to get h(x) = \( \frac{1}{\sqrt{x-1}} \).

Identify Composite Function Domain

This new function h(x) is defined only for values of x such that the expression under the square root, \( x-1 \), is positive. Thus, \( x > 1 \).

Evaluate Continuity

Lastly, check for continuity over the domain of the function. Since the function is composed of continuous functions and is defined for \( x > 1 \), h(x) is continuous for \( x > 1 \).