91Ó°ÊÓ

A composite function combines two functions into one. To form a composite function, we apply one function to the results of another function. For example, if we have two functions, \(f(x)\) and \(g(x)\), the composite function \((f \thinspace \text{g})(x)\) is found by applying \(g(x)\) first and then applying \(f\) to the result of \(g(x)\). This is written as \((f \thinspace \text{g})(x) = f(g(x))\).

In the given exercise, we use the notation \((f-g)(x)\). This isn't a true composite function in the traditional sense but rather signifies the difference between two functions. The operation here is subtraction: \((f-g)(x) = f(x) - g(x)\). Understanding this notation helps in both working with and combining various functions.

For our specific problem, we subtract the function \(g(x)\) from \(f(x)\), which involves combining the two mathematical expressions \(f(x) = x^2 - 3\) and \(g(x) = 2x + 1\).

Simplification is the process of combining like terms to make an expression easier to work with. Once we have the form of our expression, we need to simplify it to make further calculations more straightforward.

In our problem, after finding the composite expression \((f-g)(x) = (x^2 - 3) - (2x + 1)\), we distribute the negative sign and combine like terms:

• First, distribute the negative sign: \((f-g)(x) = x^2 - 3 - 2x - 1\).
• Next, combine the constant terms: \(-3\) and \(-1\) to get \(-4\).

After these steps, our simplified function becomes \((f-g)(x) = x^2 - 2x - 4\). This step is crucial for making accurate evaluations and further operations easier.