Find the compositions f ◦ g and g ◦ f, and identify their respective domains. f (x) = x² + 9, g(x) = cos x A) (f ◦ g)(x) = cos(x² + 9), 0

Question

Find the compositions f ◦ g and g ◦ f, and identify their respective domains. f (x) = x² + 9, g(x) = cos x A) (f ◦ g)(x) = cos(x² + 9), 0 < x (g ◦ f)(x) = cos² x + 9, 8 < x < 10 B) (f ◦ g)(x) = cos(x² + 9), all reals (g ◦ f)(x) = cos² x + 9, all reals C) (f ◦ g)(x) = cos² x + 9, all reals (g ◦ f)(x) = cos(x² + 9), all reals D) (f ◦ g)(x) = cos² x + 9, 8 < x < 10 (g ◦ f)(x) = cos(x² + 9), 0 < x

Quizplus · Accepted Answer

To find the composition of two functions, we substitute the inner function into the outer function.

So,

(f◦g)(x) = f(g(x)) = f(cos x) = cos^2(x) + 9

and

(g◦f)(x) = g(f(x)) = g(x^2 + 9) = cos(x^2 + 9)

The domain of (f◦g)(x) is all real numbers since cosine is defined for all real numbers, and the domain of (g◦f)(x) is also all real numbers since x^2 + 9 is always non-negative.

Therefore, the best choice is C.

Find the Compositions F ◦ G and G ◦ F