Question

Why is \[f(x)=\sin(x^2)\] continuous for all real \[x\]?

Options

• \[f(x)=(g\circ h)(x)\], where \[g(x)=\sin x\] and \[h(x)=x^2\] are continuous functions.

• Because \[\sin x\] is a rational function.

• Because \[\sin(x^2)=\tan x\] for all real \[x\].

• Because \[x^2\] is never equal to zero.

Answer 1

The function \[h(x)=x^2\] is continuous, and \[g(x)=\sin x\] is continuous for all real inputs. Since \[f(x)=g(h(x))\], the composition is continuous for all real \[x\].