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When we look at the rate at which a function is increasing, we're essentially examining how quickly the function's value rises as we move along the x-axis. An 'increasing function rate' implies that for any two points on the graph, if the second point is to the right of the first, its function value will be higher.

The concept of a function whose derivative exceeds 1 at every point suggests that not only is the function increasing everywhere, but it is doing so at a pace that is consistently brisk. The derivative—being greater than 1—indicates that for each unit we move horizontally, the function ascends by more than one unit vertically. For students envisioning this, think about climbing a hill where each step forward also takes you more than one step up; this kind of slope would be unyielding and relentless. This characteristic ensures the absence of flat spots (local minima or maxima) since the function never stops or slows down; it's always on the up.

An exponential growth curve is one of the most powerful illustrations of incessant increase. It's a specific type of curve where the rate of growth is proportional to its current value, resulting in the function accelerating upwards as it progresses along. Though the function in our exercise isn't necessarily exponential, it shares a key trait: a perpetually increasing rate.

In an exponential curve, the pace of increase is not just steady; it actually multiplies over equal intervals. To imagine this, think of it as a population of rabbits where each new generation is double the size of the last. Before you know it, you're overwhelmed by rabbits! The steep and continuous ascent in an exponential function occurs because the 'rate of change' (the derivative) is itself increasing, a concept that is echoed in our exercise's graph, though with a constant lower bound instead of an increasing derivative.

The 'function properties' hint at the inherent behavior of a function documented by its graphical representation. A regular review of a function includes its domain (where it's defined), range (the values it can take), as well as its continuity, and differentiability. For a function whose derivative is always greater than 1, some of its core traits would include being continuous (no breaks in the graph), differentiable (able to find a tangent at every point), and having no local extrema, since the graph is perpetually climbing.

Understanding these properties helps students predict the behavior of a function beyond just the points they've calculated. It's like having a road map that tells you not only about the roads you're on but also about the terrain ahead. For instance, knowing there are no local maxima or minima indicates the journey on this function's graph will only have uphill segments, with no peaks to reach or valleys to descend into.