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1. The function composition exists an operation ” ∘ ” that brings two functions f and g, and has a function h = g ∘ f such that h(x) = g(f(x)).

   Let the functions be f(x) = 4x² + 1 and g(x) = x² – 3

   The correct answer is (f o g)(x) = 4x⁴ – 96x + 37 and

   (g o f)(x) = 16x⁴ + 8x² – 2.

   What is composition function?

   The function composition exists an operation ” ∘ ” that brings two functions f and g, and has a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g exists used for the outcome of applying the function f to x.

   Given:

   f(x) = 4x² + 1 and g(x) = x² – 3

   a) (f o g)(x) = f[g(x)]

   f[g(x)] = 4(x² – 3)² + 1

   substitute the value of g(x) in the above equation, and we get

       = 4(x⁴ – 24x + 9) + 1

   simplifying the above equation

       = 4x⁴ – 96x + 36 + 1

       = 4x⁴ – 96x + 37

   (f o g)(x) = 4x⁴ – 96x + 37

   b) (g o f)(x) = g[f(x)]

   substitute the value of g(x) in the above equation, and we get

   g[f(x)] = (4x² + 1)²- 3

         = 16x⁴ + 8x² + 1 – 3

   simplifying the above equation

         = 16x⁴ + 8x² – 2

   (g o f)(x) = 16x⁴ + 8x² – 2.

   Therefore, the correct answer is (f o g)(x) = 4x⁴ – 96x + 37 and

   (g o f)(x) = 16x⁴ + 8x² – 2.

   To learn more about the function refer to:

   https://brainly.com/question/26709985

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