Problem 55 - Precalculus with Limits

Question

In Exercises 47-56, (a) use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant, and (b) make a table of values to verify whether the function is increasing, decreasing, or constant over the intervals you identified in part (a). \(f(x) = x^\frac{3}{2}\)

Step-by-Step Solution

Answer

The function \(f(x) = x^{\frac{3}{2}}\) is increasing over the range \(x \geq 0\).

Step 1: Graphing the function

Graph the function \(f(x) = x^\frac{3}{2}\) using a graphing utility. This function is the square root of a cubic function, and so, the plot will start from the origin (0,0) and will increase as x increases.

Step 2: Determining intervals visually

After graphing the function, visually identify the intervals where the function is increasing or decreasing. For this function, it can observed that the function is increasing over the entire domain \(x \geq 0\). There are no intervals where the function is decreasing or constant.

Step 3: Creating a table of values

Create a table of values to confirm the visual observations. Pick some values for x and compute the corresponding values of f(x). For instance, choose x = 0, 1, 2, 3. The corresponding f(x) values will be 0, 1, 2.83, 5.2, respectively. You can see that as x increases, f(x) also increases, which means that the function is indeed increasing over the entire domain.

Key Concepts

Increasing and Decreasing Functions

To understand increasing and decreasing functions, let's start by exploring what it means for a function to be increasing or decreasing over an interval.
An increasing function is one in which the output, or the value of the function, grows as the input increases.
Conversely, a decreasing function shows a decrease in the output as the input increases.
For the function given, \[ f(x) = x^{\frac{3}{2}} \] you will notice that it is expressed as the square root of a cubic term.
This function visually suggests an upward slope starting from the origin (0,0), indicating that as the value of \( x \) rises, \( f(x) \) does too.
Since there is no negative slope or horizontal line on the partially graphed function, it's clear that there are no decreasing or constant sections within its domain.
Therefore, for \( x \geq 0 \), this function continuously rises, making it an increasing function over its entire domain.
This concept is fundamental when graphing and analyzing any function to determine its behavior across its domain.

Table of Values

Creating a table of values provides a numerical verification of whether a function is increasing, decreasing, or constant.
This step is particularly useful after visually analyzing the graph to ensure that the visual assessment aligns with numerical data.
To construct a table of values for the function \[ f(x) = x^{\frac{3}{2}} \] select some values of \( x \) and calculate the corresponding \( f(x) \).
For example:

• \( x = 0 \) gives \( f(x) = 0^{\frac{3}{2}} = 0 \)
• \( x = 1 \) gives \( f(x) = 1^{\frac{3}{2}} = 1 \)
• \( x = 2 \) gives \( f(x) \approx 2.83 \)
• \( x = 3 \) gives \( f(x) \approx 5.2 \)

By inspecting these values, you can confirm that \( f(x) \) increases as \( x \) increases.
This tabular approach corroborates the visual findings that the function is strictly increasing over \( x \geq 0 \).
Tables like these are not just helpful for confirming assumptions but also for providing insights into the function's behavior for specific values.

Interval Identification

Identifying intervals over which functions increase, decrease, or remain constant is a crucial aspect of analyzing graph functions.
Intervals can help us understand the behavior or trend of a function and its changes over the domain.
For the function \[ f(x) = x^{\frac{3}{2}} \] we look at the function's behavior as \( x \) moves from one value to another.
Since we've determined from visual analysis and a table of values that \( f(x) \) is increasing for \( x \geq 0 \), it means we should identify the interval where this holds.

• The increasing interval can be described as \([0, \infty)\) since no maxima or minima interrupt the rising trend of the function.

Understanding the complete characterization of these intervals is important, as it provides a detailed picture of how the function behaves across its entire domain.
This information is vital when applying this knowledge to solve real-world problems or performing further mathematical operations involving the function.