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Understanding whether a function is increasing or decreasing is vital in precalculus because it tells us the behavior of the function over a specific range. To determine if a function is increasing or decreasing, observe how the function changes as the input, usually denoted by \( x \), changes.
If the function's output, denoted as \( g(x) \), increases when \( x \) increases, then the function is said to be increasing. On the other hand, if the function's output decreases as \( x \) increases, the function is decreasing. This concept can be visualized on a graph, where an increasing function will rise as you move from left to right, while a decreasing function will fall.
In the given exercise, as \( x \) increases from 1 to 5, \( g(x) \) decreases from 90 to 70; hence the function is decreasing. Observing this pattern can help in predicting the behavior of the function across its domain.

Concavity in functions gives insight into the direction of bending in the function's graph. This is important, as it helps in understanding the nature of turning points and the overall shape of the graph. A function is considered concave up if it bends upwards like a cup, and concave down if it bends downwards like a frown.
To determine concavity from a table of values, we look at how the differences between successive outputs change. If these differences increase, the function is concave up. Conversely, if they decrease, the function is concave down.
For the function in our exercise, the successive differences \( g(2) - g(1) = -10 \), \( g(3) - g(2) = -5 \), \( g(4) - g(3) = -3 \), and \( g(5) - g(4) = -2 \) are increasing. This pattern indicates that although the function is decreasing, the reduction is happening at a slower rate, thus the function is concave up.

The rate of change is a crucial concept in understanding how functions behave. It describes how one quantity changes in relation to another. In precalculus, the rate of change often refers to the change in the function's output as the input changes.
To calculate the rate of change, we take the difference in \( g(x) \) values and divide by the difference in \( x \) values for two successive points. For example, the rate of change from \( x = 1 \) to \( x = 2 \) is \( \frac{g(2) - g(1)}{2-1} = -10 \). Such calculations can reveal if the function's slope is steep or shallow, which is indicated by the value of the rate of change.
Important points to note:

• A constant positive rate signifies a consistently increasing function.
• A constant negative rate signifies a consistently decreasing function.
• Varying rates can indicate changing concavity, as seen when rates become less negative or more positive, it points towards a concave up behavior, as is the case in our exercise.

This clear understanding aids in the graphical depiction and the analytic understanding of functions.