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Adding functions is similar to adding numbers. When given two functions, like \(f(x)\) and \(g(x)\), function addition refers to creating a new function, \((f+g)(x)\). This is done by summing the outputs of \(f(x)\) and \(g(x)\) for each value of \(x\). Put simply, you are combining the effects of two functions into one.

For example, if \(f(x) = 3x\) and \(g(x) = 4x\), adding them means calculating \((f+g)(x) = f(x) + g(x) = 3x + 4x\). By simplifying this, you get \((f+g)(x) = 7x\).

This new function embodies the influence of both \(f\) and \(g\) together. It's a straightforward process when dealing with linear functions, which often results in another linear function.

The domain of a function is an essential concept. It answers the question: "For which values of \(x\) is this function defined?" This means determining the set of possible input values that can be plugged into the function without causing any issues, like division by zero or taking the square root of a negative number.

For linear functions like \(f(x) = 3x\) and \(g(x) = 4x\), the domain is quite straightforward because they are defined for all real numbers. They don't have any restrictions, so their domain is the whole set of real numbers, represented as \(\mathbb{R}\).

Consequently, when you add two linear functions, such as \(f+g\), the resulting function, \((f+g)(x) = 7x\), also has the domain of all real numbers. It's a continuous function without breaks or undefined points.

Linear functions are defined by a relationship where changes in \(x\) result in proportional changes in \(y\). Mathematically, they are represented as \(f(x) = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.

The slope \(m\) tells us how steep the line is, and it indicates how much \(y\) changes for a unit change in \(x\). The intercept \(b\) indicates where the line crosses the y-axis. In cases like \(f(x) = 3x\) and \(g(x) = 4x\), these are linear functions without a \(b\) term, meaning they cross the origin (0,0).

These functions create straight lines when graphed, indicating a constant rate of change. Understanding linear functions is crucial because they help create simple models of relationships, making them foundational in algebra and essential for grasping more complex mathematical concepts.