What is the domain of the composite function of f(x)=x^2 and g(x)=sin x when (g o f)(x) and (f o g)(x)??

when i joined the two function it looks the same to me..

(g o f)(x)

= g (f(x) )
= g(x^2)
= sin (x^2)

domain: any real x

(f o g)(x)
= f (g(x) )
= f(sinx)
= (sinx)^2 or sin^2 x

domain: any x

thank you!

To find the domain of the composite function (g o f)(x), which means applying g to f(x), we first need to consider the individual domains of the functions f(x) and g(x).

The function f(x) = x^2 is a polynomial function, and polynomials are defined for all real numbers. Therefore, the domain of f(x) is all real numbers.

On the other hand, the function g(x) = sin(x) is a trigonometric function, particularly the sine function, which is also defined for all real numbers. So, the domain of g(x) is also all real numbers.

Now, to determine the domain of the composite function (g o f)(x), we need to look at the input values that are valid for both functions when applied in sequence. Since both f(x) and g(x) have a domain of all real numbers, we can conclude that the domain of (g o f)(x) is also all real numbers.

The composite function (f o g)(x) implies applying f to g(x). Again, both f(x) and g(x) have a domain of all real numbers. Therefore, the domain of (f o g)(x) is also all real numbers.

In summary, the domains of both composite functions (g o f)(x) and (f o g)(x) are all real numbers.