If f(x) = sin(x) and g(x) = cos(x), how do you find g(f(x)) and f(g(x))?

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Finding g(f(x)) and f(g(x))

To solve the problem, we need to find two compositions of functions:

  1. g(f(x))
  2. f(g(x))

Step 1: Finding g(f(x))

Given:

  • f(x) = sin(x)
  • g(x) = cos(x)

To find g(f(x)), we substitute f(x) into g(x):

g(f(x)) = g(sin(x)) = cos(sin(x))

Final Result for g(f(x))

Thus, g(f(x)) = cos(sin(x)).

Step 2: Finding f(g(x))

Next, we look for f(g(x)). Here, we substitute g(x) into f(x):

f(g(x)) = f(cos(x)) = sin(cos(x))

Final Result for f(g(x))

Therefore, f(g(x)) = sin(cos(x)).

Summary

In conclusion, we have:

  • g(f(x)) = cos(sin(x))
  • f(g(x)) = sin(cos(x))

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