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A constant function is one of the simplest forms of function you will encounter. It is characterized by a constant value for all inputs within a specified domain. In other words, no matter what value of \( x \) you choose within a specific range, the output or \( y \)-value will always remain the same.
This can be visualized as a horizontal line on a graph. Let's take an example used in the problem: \( P(x) = \frac{1}{8} \). For this function, \( y \) is always \( \frac{1}{8} \) for any value of \( x \) within the interval where the function is defined.
Understanding that the value doesn't change, regardless of \( x \), simplifies many calculations, particularly when determining the area under the curve.

The interval of interest refers to the particular segment of the \( x \)-axis over which we're considering the function's behavior. In the given exercise, we're focusing on the interval \( [2, 5] \). This means we are interested in observing how the function behaves only between \( x = 2 \) and \( x = 5 \).
Defining an interval is crucial because it determines the limits for any analysis like area calculation. It's essentially setting boundaries for your math problem. Outside of this interval, the values of the function might be different, or as in our example problem, nonexistent.

• Inside the interval \( [2, 5] \), \( P(x) = \frac{1}{8} \).
• Outside this interval, the function is zero or undefined.

Identifying and sticking to this interval simplifies finding the total area under the function.

Calculation of the area under a curve is an essential concept in calculus often applied when using integrals. However, for a constant function like \( P(x) = \frac{1}{8} \), it can be simplified as the area of a rectangle. The formula to find this area is straightforward: multiply the height of the rectangle by its width.
Here, **Height** is \( \frac{1}{8} \), and **Width** is \( 5 - 2 = 3 \). To find the area under the function from \( x = 2 \) to \( x = 5 \), you simply calculate:

• Area = Height x Width
• Area = \( \frac{1}{8} \times 3 \)
• Area = \( \frac{3}{8} \)

Thus, the area under the constant function over the interval [2, 5] is \( \frac{3}{8} \), representing the shaded region on the graph.

Graphing functions provides a visual representation of how a function behaves over its domain. For constant functions like \( P(x) = \frac{1}{8} \), the graph is a simple horizontal line at \( y = \frac{1}{8} \) over their interval of definition.
When you graph \( P(x) \) from \( x = 2 \) to \( x = 5 \), it results in a straight, flat line parallel to the \( x \)-axis. This line clearly demonstrates that regardless of the \( x \)-value within this interval, the \( y \)-value remains constant.
To reinforce, the graph for the problem is simply a flat line:

• The line starts at \( x = 2 \) and ends at \( x = 5 \).
• It shows \( y = \frac{1}{8} \) consistently throughout this interval.

Graphing helps affirm our calculated area visually, with the area under this line representing the constant value kept throughout the interval.