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When dealing with functions, one of the first steps is learning how to evaluate them. This simply means finding the value of the function for a specific input. Let's break down the process:

Given functions like:
\( f(x) = x^2 - 9 \), \( g(x) = 2x \), and \( h(x) = x - 3 \),
you can plug in any specific value for \(x\) to find the corresponding output. For instance, to evaluate \(f(x)\) at \(x = -3\), substitute -3 into the equation:
\[ f(-3) = (-3)^2 - 9 = 9 - 9 = 0 \]
This way, we find that \(f(-3) = 0\).
Evaluating functions is fundamental in understanding how they behave and is often the first step in solving more complex problems.

Subtracting functions is an operation that combines two functions by subtracting the value of one from the value of the other. Let’s look at an example using our given functions, \(f(x)\) and \(g(x)\).

To find \((f - g)(x)\), you compute:
\[ (f - g)(x) = f(x) - g(x) \]
This means you subtract the output of \(g(x)\) from the output of \(f(x)\).

For instance, finding \((f - g)(-3)\) involves these steps:

• Evaluate \(f(x)\) at \(x = -3\):
  \[ f(-3) = (-3)^2 - 9 = 0 \]
• Evaluate \(g(x)\) at \(x = -3\):
  \[ g(-3) = 2(-3) = -6 \]
• Subtract:
  \[ (f - g)(-3) = f(-3) - g(-3) = 0 - (-6) = 6 \]

So, \((f - g)(-3)\) results in 6. This method is essential when comparing the behaviors of different functions or when simplifying combined function expressions.

Composite functions involve combining two functions into a single function, wherein the output of one function becomes the input of another. Consider two functions \(f(x)\) and \(g(x)\). The composite function \((f \circ g)(x)\) means you first apply \(g(x)\), and then apply \(f\) to the result of \(g(x)\).

Mathematically, it looks like this:
\[ (f \circ g)(x) = f(g(x)) \]

Let's illustrate this with an example:
For \(f(x) = x^2 - 9\) and \(g(x) = 2x\), find \((f \circ g)(x)\).

You start by finding \(g(x)\):
\[ g(x) = 2x \]
Next, you use this expression as the input for \(f(x)\):
\[ f(g(x)) = f(2x) = (2x)^2 - 9 = 4x^2 - 9 \]

So, \((f \circ g)(x)\) simplifies to \(4x^2 - 9\).
Understanding composite functions is crucial for tackling more advanced mathematical topics and for understanding how different functions can interact with each other.