Question

Discuss the continuity of the composite function \(h(x)=f(g(x))\). $$ \begin{array}{l} f(x)=\sin x \\ g(x)=x^{2} \end{array} $$

Short answer

The composite function \(h(x) = f(g(x)) = f(x^2) = \sin(x^2)\) is continuous for all real numbers as both \(f(x)\) and \(g(x)\) are continuous for all real numbers.

Step-by-step solution

Discussion of continuity for function g

Look at function \(g(x) = x^2\). This is a basic quadratic function, and such functions are continuous for all real numbers. Therefore, \(g(x)\) is continuous everywhere.

Discussion of continuity for function f

Now look at function \(f(x) = \sin x\). This function is also continuous for all real numbers.

Discussion of continuity for composite function h

The composite function \(h(x) = f(g(x)) = f(x^2)\) is the application of continuous function \(f\) at \(x^2\). Since \(x^2\) is continuous for all real numbers (from step 1), and \(f(x)\) is also continuous for all real numbers (from step 2), the composite function \(h(x)\) will also be continuous for all real numbers. In other words, composite function of two continuous functions is also a continuous function.