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When it comes to functions, there’s more to them than just plugging in numbers and getting results. Functions can be manipulated and combined in various ways to create new functions, similar to arithmetic operations with numbers. One such manipulation is the addition of functions.

Let's imagine we have two functions, like our two friends, 'Function f' and 'Function g'. These functions each take a number, transform it in their own unique way, and spit out a result. If we want to create a new function that combines the transformation powers of both 'f' and 'g', we can add them together. This combination, known as the addition of functions, is straightforward: we simply add the outputs of 'f' and 'g' for the same input value.

• The addition of functions is noted as \((f + g)(x) = f(x) + g(x)\),and it means that for each input 'x', the output is the sum of the outputs from 'f' and 'g'.
• This operation requires that 'f' and 'g' are defined at 'x'.
• The result is a new function that combines the contributions of both 'f' and 'g'.

But, the magic of functions doesn’t end with addition. They can also be transformed one after another through another process called composition.

Imagine you have a machine that paints objects red and another that shapes objects into spheres. If you used the sphere machine first and then the painting machine, you’d get a red sphere. This sequential process is similar to how we compose functions in mathematics.

Function composition involves taking two functions, where the output of one function becomes the input to the other. It’s like a relay race where one function passes the baton to the next. If we denote 'Function f' as the final action and 'Function h' as the first action, then the composition is written as \((f \circ h)(x) = f(h(x))\).

• Here \(f \circ h\) is the composed function, and 'x' is the starting value.
• We first apply 'h' to 'x', then apply 'f' to the result of \(h(x)\).
• It's important to note that the output of 'h' must be a suitable input for 'f'.

When we are confronted with a compound machine, or in our case, a composition of functions like \((f+g) \circ h\), we can apply the rules of composition to uncover its operation, layer by layer, as the next section will show.

Let's return to our example of functions 'f' and 'g' and see what happens when they join forces, both being followed by 'Function h'. We want to understand what happens when we add these functions first and then compose with 'h', compared to when we compose them with 'h' individually and then add them.

Following the chain of operations, we begin by adding 'f' and 'g' together to get a new function, \((f + g)\), and then we compose this with 'h'. As per the rules of composition, this yields \(((f+g) \circ h)(x) = f(h(x)) + g(h(x))\). Now, if we go the other route by composing first and then adding, we get two separate compositions, \(f \circ h\) and \(g \circ h\), and their sum would be \((f \circ h)(x) + (g \circ h)(x)\).The intriguing aspect here is that both processes – adding first then composing, versus composing first then adding – give us the same result! This leads to the conclusion that

• Combining functions through addition and then composing with another function is interchangeable with composing each function individually and then adding them together.
• Thus, the process shows a property of functions known as distributivity of composition over addition.
• This property is not just mathematically elegant but it’s also practically useful, simplifying complex function operations.

This conclusion reinforces the concept of functions as dynamic mathematical objects that can be modified, reshaped, and combined in a multitude of ways to reveal new properties and results.