Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the function and (b) determine the open intervals on which the function is increasing, decreasing, or constant. $$f(x)=3$$

Step-by-Step Solution

Answer

The function \(f(x) = 3\) is graphed as a horizontal line intersecting the y-axis at 3. As it is a constant function, it is neither increasing nor decreasing for all real numbers. It is constant over the interval \(-\infty, +\infty\).

Step 1: Plotting the function \(f(x)=3\)

A constant function of value '3' will appear as a horizontal line across the y-axis at the point 3. Regardless of the x-value, the function value will always be 3.

Step 2: Determine where the function is increasing, decreasing or constant

The function \(f(x) = 3\) is a constant function, it will not increase or decrease at any point. All constant functions are neither increasing nor decreasing, they're constant regardless of the interval selected.

Key Concepts

Graphing Utility

When using a graphing utility, it can help to visualize the behavior of a function. Here, we're focusing on the function \(f(x) = 3\). A graphing utility is a tool that plots the graph of equations, showing us exactly how the function behaves at different points.Imagine plugging in several values for \(x\) into the function \(f(x) = 3\). Regardless of what value you choose for \(x\), the output remains the same, which is 3. Using a graphing tool will depict this understanding by displaying a horizontal line that cuts through the y-axis at the point \(y = 3\). This horizontal line clearly illustrates that the function's value doesn't change. It's a straightforward way to observe the consistency and constancy of the function.

Increasing and Decreasing Functions

Understanding when a function is increasing or decreasing is crucial in calculus and algebra. When a function is increasing, as the x-values move from left to right, the y-values go up. In contrast, when a function is decreasing, the y-values go down as you move right along the x-axis.For our function \(f(x) = 3\), which creates a horizontal line on the graph:

• The y-values stay the same at 3, no matter how much we change \(x\).
• This tells us that the function isn’t increasing or decreasing—it is constant.

No slopes or changes in direction occur in this graph. This constancy means that the function does not have any increasing or decreasing intervals. The function value remains constant over its entire domain.

Constant Function Analysis

Constant functions are unique compared to other functions. They have an important characteristic: their graph is a horizontal line, and they maintain the same y-value, regardless of x.In the function \(f(x) = 3\):

• The y-value is constantly 3 for every possible value of \(x\).
• These graphs might seem simple, but they play a critical role in mathematics, representing scenarios where a quantity remains unchanged despite other variations.

If you imagine a straight road with no inclination, you’re picturing the essence of a constant function. It's level and predictable, without any ups or downs, valuing consistency above all. Constant functions thus provide a simple yet effective way to convey stability in various scenarios.