CAT Function of a function i.e. fog or gof - Practice Questions & MCQ

Function of a Function (Composite Function)

Definition:

Given two functions f and g , the composite function f ∘ g (read as "f of g") is defined as:

Similarly, g ∘ f is:

Characteristics:

- The domain of the composite function f ∘ g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.

- Composition is not commutative. This means f ∘ g is not necessarily the same as g ∘ f.

Application of Previous Concepts:

- The composite function will be even if both f and g are even, or both are odd.

- The composite function will be odd if one of the functions is odd and the other is even.

- If one of the functions has an inverse, the composition can sometimes be simplified.

Foundation Building Questions:

Question 1:

Given functions

Solution: 

Question 2:

If f(x) = |x| and g(x) = x - 4, determine if g ∘ f(x) is even, odd, or neither.

Solution: 

To determine if the function is even:

Evaluate g ∘ f (-x): |-x| - 4 = |x| - 4

Since  g ∘ f(-x) = g ∘ f(x), the function is even.

Tips and Tricks related to 'Function of a Function':

1. Order Matters: 

    - Always be mindful of the order in which functions are composed. f \circ g is not the same as g \circ f in general.

  2. Simplify When Possible: 

    - Before evaluating a composite function at a point, see if you can simplify the function.

  3. Use Previous Knowledge: 

    - Utilise your understanding of even and odd functions, modulus functions, and inverses to simplify and analyse composite functions.

  4. Domain Check: 

    - Before evaluating or working with a composite function, ensure that the values you're working with lie in the correct domain.

5. Graphical Insight: 

    - Visualising the graphs of individual functions can sometimes provide insights into the behaviour of the composite function, especially regarding its domain, range, and symmetry.

Mastering composite functions requires understanding each function's behaviour and how they affect each other. Practice and visualisation are key components in grasping the intricacies of this topic.