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In many mathematical problems, you might have to substitute one function into another. This transformation is crucial for combining functions or expressing one variable in terms of another. For instance, in our example, we had two functions: \( y = 2a - 3 \) and \( a = 3x + 1 \). Notice that 'a' is an intermediate variable between 'y' and 'x'. By substituting \( a \) in the first equation with its expression from the second equation \( a = 3x + 1 \), you bridge 'y' directly with 'x'. This is done in the following manner: replace every 'a' in \( y = 2a - 3 \) with \( 3x + 1 \). Hence, we get: \[ y = 2(3x + 1) - 3 \]. This step of substituting functions acts like weaving a thread through two separate pieces, making them part of a single whole.

After substituting the function, the resulting equation often needs simplification. Simplifying an equation makes it more manageable and reveals the direct relationship between the variables. For our problem, we substituted \( a = 3x + 1 \) into \( y = 2a - 3 \), resulting in \( y = 2(3x + 1) - 3 \). The next step is to simplify this. We distribute the 2 across the terms inside the parentheses: \[ y = 2 \times 3x + 2 \times 1 - 3 \ y = 6x + 2 - 3 \]. Now, combining the constants (2 and -3) gives: \[ y = 6x - 1 \]. This final equation is much simpler and directly shows that 'y' is a function of 'x'.

Function composition involves creating a new function by combining two or more functions. In terms of understanding how one quantity changes with respect to another, function composition is very useful. For this exercise, the two functions can be described as follows: function 1: \( a = g(x) = 3x + 1 \) and function 2: \( y = f(a) = 2a - 3 \). The objective here was to find \( y \) as a function of \( x \), denoted typically as \( y = f(g(x)) \). To achieve this, plug the output of function 1 (\( g(x) \)) into function 2 (\( f(a) \)). In concrete terms, we compute \( f(g(x)) \) to find \( y = f(g(x)) \): \[ y = f(g(x)) = 2(3x + 1) - 3 \ y = 6x - 1 \]. Thus, \( y \) is effectively expressed as a direct function of \( x \). This shows how intricate functional relationships can be merged into a single, more straightforward function.