Problem 27 - Algebra and Trigonometry Real Mathematics, Real People

Question

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 graph of function \(f(x)=3\) is a straight horizontal line passing through the point on the y-axis at y=3. The function does not increase or decrease at any point in the domain. It is constant over the interval \( -\infty < x < \infty\).

Step 1: Graphing the function

Using a graphing tool of your choice, plot the equation \(f(x) = 3\). Whatever the x-coordinate is, the y-coordinate will always be 3 since the function is constant. As a result you should see a straight horizontal line passing through the point (0,3) on the y axis.

Step 2: Checking for intervals of increase or decrease

An increase or decrease in a function involves a change in the y-coordinate as the x-coordinate changes. But in this case, as we move along the x-axis, the y-coordinate remains the same (3). So there are no points on the graph where the function \(f(x)\) is increasing or decreasing. It is constant everywhere.

Step 3: Checking for intervals of constancy

A function is considered constant on an interval if the function has the same value for any number in that interval. As we can see from the graph and the formula of this function, for every value of x, the function value is always 3. So the function \(f(x) = 3\) is constant over the entire domain, \( -\infty < x < \infty\).

Key Concepts

graphing utilities

Graphing utilities are incredibly helpful tools in visualizing functions, especially when dealing with simple or complex mathematical functions. For the constant function \( f(x) = 3 \), a graphing utility allows us to instantly see that the graph is a horizontal line. This line runs parallel to the x-axis and crosses the y-axis at the point \((0, 3)\).

Using a graphing tool not only simplifies the task of plotting functions but also enhances understanding by providing a clear visual representation. Most modern graphing utilities enable easy manipulation of scales and offer features like zoom, which can make it easier to analyze different aspects of the graph. As you get more comfortable using these tools, they can become invaluable in your study of functions and calculus.

For learning students, graphing utilities illustrate that a constant function, like \( f(x) = 3 \), doesn't depend on any individual x-value, since it maintains the same output (3) universally across the x-domain.

function intervals

Understanding function intervals is key in determining where a function behaves in certain ways. Intervals are the x-values where the function either increases, decreases, or remains constant.

For the function \( f(x) = 3 \), identifying intervals is straightforward. This function is constant because no matter what value x takes on the real number line, the y-value remains at 3. Therefore, it doesn’t increase or decrease; it's unchanging over any interval. The interval of constancy for this function is the entire set of real numbers, represented by \(-\infty < x < \infty\).

In contrast, a non-constant function might have specific intervals where it increases or decreases. Recognizing these intervals can provide insights into the behavior of more complex functions, whereas for a constant function, the interval of constant behavior is simply the whole domain.

horizontal line

A horizontal line on a graph is a line that runs left to right and is parallel to the x-axis. It is characterized by having the same y-value at every point. For example, when graphing \( f(x) = 3 \), the line is horizontal and passes through all points where the y-coordinate is 3 regardless of x.

This kind of line visually represents a constant function, meaning there’s constant value output as input changes. No matter how far you move along the x-axis, the graph stays flat, indicating no increase or decrease. It's an easy way to see that the function's output doesn’t change.

Additionally, graphing a horizontal line can help one understand key properties of constant functions in real-world applications. It's a fantastic demonstration of stability or equilibrium in various contexts, such as economics or physics, where a quantity doesn’t vary over time or distance.