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Function evaluation is the process of finding the value of a function for a specific input. For example, if you have a function defined as \(f(x) = x^2 - 9\) and you wish to evaluate it at \(x = 3\), you simply substitute 3 in place of \(x\). You get:
\(f(3) = 3^2 - 9 = 9 - 9 = 0\).
Function evaluation helps us understand how a function behaves for different input values.
If you can break down a problem into simpler steps and practice substituting values, function evaluation will become easier over time.

When dealing with compositions of functions, it’s important to identify the inner and outer functions. These terms help you understand the order in which you should evaluate the functions.

• Inner function: The function that you evaluate first. For example, in the composition \(f(g(x))\), \(g(x)\) is the inner function.
• Outer function: The function you apply after evaluating the inner function. In this case, \(f(x)\) is the outer function.

Recognizing the inner and outer functions is crucial for correctly solving compositions. If you mix up the order, you may get the wrong result. Always remember to start with the inner function and then move to the outer function.

Substitution in functions involves replacing the variable in a function with a specific value or another expression. This is a fundamental technique in working with functions, especially in compositions. Let's use the functions from our original problem as an example:

Suppose we have \(h(x) = x - 3\) and we want to find \(h(-1)\). To do this, substitute \(-1\) for \(x\) in \(h(x)\):
\(h(-1) = -1 - 3 = -4\)

Now we need to use this result in another function, say \(f(x) = x^2 - 9\). We substitute \(-4\) (the result from \(h(-1)\)) into \(f(x)\):
\(f(-4) = (-4)^2 - 9 = 16 - 9 = 7\).

By practicing this substitution process, you can handle even more complex compositions of functions with ease. Breaking down the steps and substituting methodically ensures you get accurate results.