Problem 54 - 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). \(f(x) = \sqrt[x]{x+3}\)

Step-by-Step Solution

Answer

Use a graphing utility to graph \(f(x) = \sqrt[x]{x+3}\), identify the intervals of increase, decrease, and constant, and then make a table of values to verify these intervals.

Step 1: Plotting the Graph

Start by using a graphing utility to plot \(f(x) = \sqrt[x]{x+3}\). Observe the graph to identify the regions where the function increases, decreases, or remains constant.

Step 2: Identifying Intervals

From the graph, identify the intervals where the function is increasing, decreasing, or constant. These are visible as segments of the graph where the curve is rising (increasing), falling (decreasing), or flat (constant).

Step 3: Creating a Table of Values

Choose a sample of x-values within each of the identified intervals. These do not need to be specific; they are only to verify the behavior observed in the graph. Compute the corresponding y-values for each of these x-values.

Step 4: Verification

Compare the y-values within each interval. If y-values are increasing with x, then the function is increasing on that interval. If y-values are decreasing with x, then the function is decreasing on that interval. If y-values are constant with varying x, then the function is constant on that interval. Verify whether these observations match with the intervals identified in Step 2.

Key Concepts

Increasing and Decreasing Intervals

When we look at the graph of a function, noticing whether it is going up, down, or staying flat is key to understanding its behavior.
When we say that a function is "increasing," we mean that as you move from left to right on the graph, the function's output (or y-value) goes up. Picture a hill that you're walking up. On the other hand, a "decreasing" function looks like a downhill slope where, as you move right, the outputs decrease.
Sometimes, parts of the graph can look horizontal; these are intervals where the function is "constant." That means no matter which point you choose in this segment, the y-value doesn't change.
To identify these intervals, you may need to use the graph of the function. Spot areas where the curve rises, and mark these as increasing. Do the same for where it falls or stays flat. Each interval gives insight into how the function behaves across different portions of its domain.

Table of Values

A table of values is a practical tool for verifying changes in a graph. By selecting different x-values, you can calculate their corresponding y-values. This process helps confirm what you've observed in a graph.
Start by choosing some x-values from each interval you determined earlier— increasing, decreasing, or constant—and record the results. Your table might look something like this:

• For increasing sections, observe that y-values get larger.
• For decreasing parts, check that y-values get smaller.
• For constant areas, check that y-values remain the same.

With this table in hand, examine how the y-values change.
It's a handy method for making sure the visual interpretation you've made from the graph aligns with numeric data. Moreover, seeing numbers can sometimes reveal changes you may not clearly see in the graph.

Graphing Utility

Graphing utilities, like scientific calculators or computer programs, are essential for visualizing functions. They save time, reduce manual plotting errors, and offer a precise representation of complex functions.
To use a graphing utility for a function like \(f(x) = \sqrt[x]{x+3}\), input the equation and let the tool graph it. Immediately, you gain a visual insight into where the function rises, falls, or stays level.
Modern graphing utilities often allow zooming in on particular sections for a closer look, and some even calculate the derivative to offer more precise data on increasing or decreasing intervals.

• They simplify graph analysis by providing accurate graphs.
• Help identify key features, such as peaks and troughs.
• Offer calculations to confirm visible trends.

Relying on a graphing utility can make understanding functions much clearer and more efficient, especially for complex or unfamiliar graphs.