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When working with function composition, the first step is to evaluate the inner function. The inner function is the function closest to the variable, which in our case is denoted as \( g(x) \). Begin by substituting the given input value \( x \) into the function \( g \). This step might seem straightforward, but it's crucial for setting up the function composition correctly.

To illustrate, consider if \( g(x) = 2x + 3 \). If you need to evaluate when \( x = 4 \), substitute \( x \) with 4: \( g(4) = 2(4) + 3 = 11 \). This value serves as the input for the next function in the composition.

Remember, the inner function evaluation sets the stage for the rest of the process. It transforms the input from a simple \( x \) into something that can be used in the next level of evaluation.

Following the evaluation of the inner function, we move to the outer function evaluation. The outer function, denoted \( f \), uses the result from the inner function as its input. This is the core of function composition, where the output of one function becomes the input for another.

Suppose the outer function is \( f(x) = x^2 + 1 \), and the result from the inner function was 11 (as previously calculated). To evaluate, substitute 11 into \( f \): \( f(11) = 11^2 + 1 = 122 \).

The outer function takes on a new role, as it is not using direct inputs from outside the composition, but rather the transformed outputs from the inner function. This step effectively combines the effects of both functions into a single, composed function.

The substitution method is a foundational technique in function composition that streamlines the process of evaluating nested functions. Essentially, it boils down to using the output from one function as the input for another, achieving both simplicity and elegance in computations.

Here's a quick guide to using the substitution method effectively:

• First, independently evaluate the inner function using the given value of \( x \).
• Next, take the output from this evaluation and substitute it directly into the outer function.
• Finally, calculate the result of the outer function with this new input to obtain the complete result of the composed function.

By proceeding step-by-step, the substitution method keeps the process structured. This ensures that no matter how complex or simple the functions involved are, the same logical procedure will yield the correct composition result each time.