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When we talk about function composition, we are referring to combining two functions in such a way that the output of one function becomes the input to the other. This process creates a new function, which gives us a new rule for our inputs. Imagine having two machines, where the output of the first one feeds directly into the second. Composing functions works exactly like that.

In algebra, the notation for composing two functions, say function f and g, is written as f \(\circ\) g, which reads ' f composed with g'. It's important to remember the order matters— f \(\circ\) g is generally different from g \(\circ\) f. We evaluate f \(\circ\) g by first applying g to our input and then applying f to the result of g. The process is a fundamental concept that underscores many mathematical operations and is essential in understanding the inner workings of more advanced functions in calculus and beyond.

fog and gof are shorthand notations for expressing the composition of functions f and g. 'fog', written as f \(\circ\) g, means that you apply g first and then f. On the flip side, 'gof', or g \(\circ\) f, flips the process.

To illustrate, consider the given functions f(x) = 2x - 3 and g(x) = 2x - 3. For 'fog', we substitute g(x) into f(x), yielding a new function f(g(x)) = 2(2x - 3) - 3. For 'gof', we perform the opposite, substituting f(x) into g(x), resulting in g(f(x)) = 2(2x - 3) - 3. Interestingly, because both f and g have identical expressions in our example, both 'fog' and 'gof' yield the same result, which is a special case and not typical in composition of different functions.

Algebraic operations are the bread and butter of manipulating mathematical expressions and functions. They include addition, subtraction, multiplication, division, and the application of functions (like powers and roots). When working with functions, we also utilize these operations to combine, decompose, and transform functions.

In the context of function composition, algebraic operations refer particularly to the manipulation involved in substituting one function into another. With our functions f(x) = 2x - 3 and g(x) = 2x - 3, we performed algebraic operations by substituting and simplifying to find the resulting expressions for both 'fog' and 'gof'. These operations allowed us to see not just the individual functions but also how they interact when combined to form a composite function.

Function substitution is essentially when one function takes the place of the variable in another. This technique is at the core of function composition and can be used to solve complex equations or to find specific values of composed functions.

In the exercise, for f \(\circ\) g (\fog\fs), we replaced the x in f(x) with g(x) and then simplified. Similarly, for g \(\circ\) f (\fog\fs), we substituted f(x) for the x in g(x). These substitutions are straightforward but critical to understanding the underlying behavior of composed functions. The methodical application of function substitution is what leads to the composite function's formula, which allows for further analysis or graphing of the new function.