Determine whether the statement is true or false. If it is...

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $g \circ f$ is defined at $x=a,$ then $f \circ g$ must also be defined at $x=a$
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Key Concepts

Function Composition

Function composition is the process of applying one function to the results of another. It involves taking an input, applying a function to it, and then applying a second function to the result of the first. This requires that the output of the first function lands within the domain of the second function.

Domain Considerations

When composing functions, it is important to consider the domains of both functions. For a composition to be defined at a particular input, that input must be in the domain of the first function, and the result of that first function must fall within the domain of the second function. The reverse composition may not be defined if these conditions do not hold in the opposite order.

Non-Commutativity of Composition

Function composition is not commutative, which means that the order in which functions are composed matters. Even if one composition, such as g ? f, is defined at a certain input, the reverse composition f ? g might not be defined at that same input because the necessary domain and range conditions may not be met.

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