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When dealing with functions, sums often appear in various mathematical problems. A sum of functions is simply the combination of two or more functions into one by adding their outputs. For example, if you have two functions, say \( f(x) \) and \( g(x) \), their sum is represented as \( h(x) = f(x) + g(x) \). By adding the results of each function together for any given \( x \), you create this new function \( h(x) \).

This can be visualized easily by picturing the graphs of \( f(x) \) and \( g(x) \). The graph of the sum function \( h(x) \) will have values that are the sum of the respective y-values at each x on these graphs.

Understanding how the sum of two functions behaves is crucial because it can tell us more about the structure of these functions. Especially when each individual function has a specific property, like being increasing, the sum often retains this property too. This leads us to important discussions on function properties and proof techniques.

Functions have various properties that define how they behave. A critical property discussed in this exercise is the 'increasing function'. A function \( f(x) \) is called increasing on an interval if for any two numbers \( x_1 \) and \( x_2 \) within that interval, where \( x_1 < x_2 \), the inequality \( f(x_1) < f(x_2) \) holds true.

This property tells us that as you move along the x-axis, the output of the function only gets larger. It doesn't stay the same or decrease, but always goes up. This is intuitive if you imagine climbing a hill — your height (y-value) keeps increasing as you move forward (increase x).

Recognizing these properties helps mathematicians and students predict or determine how combinations of functions, like sums, behave. In the case of increasing functions, if \( f(x) \) and \( g(x) \) are both increasing, we conclude that the sum \( h(x) = f(x) + g(x) \) is also increasing. By understanding these properties intuitively, solving more complex function-related problems becomes easier.

To demonstrate that the sum of two increasing functions is also increasing, proof techniques are essential. In mathematics, proofs provide a logical basis for asserting claims with certainty. In this case, the technique involves showing that for any two points \( x_1 \) and \( x_2 \) (where \( x_1 < x_2 \)), the sum function \( h(x) = f(x) + g(x) \) satisfies \( h(x_1) < h(x_2) \).

The proof starts from the known properties of the increasing functions: \( f(x_1) < f(x_2) \) and \( g(x_1) < g(x_2) \). We then add these inequalities to derive \( f(x_1) + g(x_1) < f(x_2) + g(x_2) \). This inequality shows that \( h(x_1) < h(x_2) \), confirming that \( h(x) \) is increasing.

Such proofs rely on logical steps and the properties inherent to the functions. Incrementally, they build from assumed truths (like the increasing nature of \( f(x) \) and \( g(x) \)) to derive new facts about the sum function. By following this logical process, the conclusions drawn are general and trustworthy.