24.1.6. Composition of Functions

Try these first

1. 1.

   Define function composition.

   Hint

   Think about how functions interact with each other.

2. 2.

   What is required for composing functions?

   Hint

   Consider the sets involved.

3. 3.

   What is the main requirement for two functions to be composed?

   • Range of g is subset of Domain of f
   • Range of f is subset of Domain of g
   • Both must be equal

   Hint

   Think about how the functions are paired.

4. 4.

   True or False: Function compositions can be commuted.

   • True
   • False

   Hint

   Consider what you learned about the order of operations.

5. 5.

   Let f(x) = 2x, g(x) = x + 5, and h(x) = x^2. Compute h◦(f◦g)(x) and determine if the result matches (h◦f)◦g(x).

   Hint

   Calculate both sides step by step, evaluating at similar points.

6. 6.

   Propose a different pair of functions, f and g, and demonstrate that f◦g is not equal to g◦f by providing specific values.

   Hint

   Choose different simple functions to illustrate this behavior.