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Before diving into composite functions, it's crucial to understand function notation. Function notation provides a way to denote relationships between variables. For instance, if we have a function named \( f \), we can write \( f(x) \) to represent the function applied to the variable \( x \). This is read as 'f of x'. The function \( f \) encompasses a set of rules or operations that are applied to \( x \). Understanding this notation is fundamental because it will be used frequently in more advanced topics, like function composition and evaluating functions.

• \( f(x) \) - Means the function \( f \) applied to \( x \).
• The value inside the parentheses is the input.
• The result of the function is the output.

Always pay attention to the variable inside the parentheses, as it indicates the input to the function.

Now that you're familiar with function notation, let's talk about function composition. Function composition involves combining two functions to form a new function. The notation for a composite function looks like this: \( (f \,\circ\, g)(x) \), and it means you apply function \( g \) first and then apply function \( f \) to the result of \( g \). Written differently, \( (f \,\circ\, g)(x) = f(g(x)) \).
In our original exercise, we have \( (f \,\circ\, f)(4) \). This means we first find \( f(4) \), and then plug the result back into the function \( f \). Understanding this step-by-step approach is essential to solving problems involving composite functions.
Steps to solve a composite function:

• First, apply the inner function.
• Next, apply the outer function to the result of the first step.

Keep practicing to get more comfortable with these steps!

The final step in our problem-solving journey is evaluating functions. Evaluating a function means finding the output when a specific input is given. Use the table or equation provided to look up values.
Let's break it down using our exercise as an example:

• First, find \( f(4) \): According to our table, when \( x = 4 \), \( f(4) \) is 3.
• Next, find \( f(3) \): Using the table again, when \( x = 3 \), \( f(3) \) is 1.

This means that \( f(f(4)) = 1 \).
When evaluating functions, always ensure you're using the correct table or equation, and double-check your inputs. This attention to detail will help you avoid errors and improve your problem-solving skills.