How do you find fg when given the functions fx = x^2 + 16 and gx = x + 4, and what is the domain of fg?

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To find fg, we need to substitute the function g(x) into f(x). Here, we have:

  • f(x) = x² + 16
  • g(x) = x + 4

Now we will compute f(g(x)):

f(g(x)) = f(x + 4) = (x + 4)² + 16

Next, we expand the expression:

(x + 4)² = x² + 8x + 16

Thus, adding 16 to this result:

f(g(x)) = x² + 8x + 16 + 16 = x² + 8x + 32

So, we find that:

fg = x² + 8x + 32

Now, let’s discuss the domain of fg. Since both f(x) and g(x) are polynomial functions, their domains are all real numbers. Therefore, the composition fg is also defined for all real numbers.

In conclusion:

  • fg = x² + 8x + 32
  • Domain of fg: All real numbers (−∞, +∞)

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