Problem 32 - 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. $$f(x)=x \sqrt{3-x}$$

Step-by-Step Solution

Answer

After graphing the function \(f(x)=x \sqrt{3-x}\), it's noticeable that the function increases on the interval \(x \in [0,3]\). There are no intervals on which the function decreases or remains constant.

Step 1: Graphing the function

To graph the function \(f(x)=x \sqrt{3-x}\), it can be plotted using a graphing utility. The x-axis will represent the 'x' values, and the y-axis will represent the 'f(x)' values. For every x-coordinate on the x-axis, calculate the corresponding y-coordinate which is the value of \(f(x)\) at that point. The trailing sign change in the function \(x \sqrt{3-x}\) at the zeros \(x=0\) and \(x=3\) needs to be observed.

Step 2: Determine the domain

The domain of \(f(x)=x \sqrt{3-x}\) is derived from the conditions for which the square root is defined. Since you can’t take the square root of negative numbers and the x term multiplies on it, the domain is determined as \(x \in [0,3]\).

Step 3: Identify intervals of increase and decrease

By observing the function graphed visually, determine where the slope of the function is positive or negative. When \(x \in [0,3]\), the function is increasing, as the slope on these intervals is positive.

Step 4: Identifying constant intervals

If there are any intervals where the function neither increases nor decreases the function is constant. In the case of \(f(x)=x \sqrt{3-x}\), there are no such intervals.

Key Concepts

Intervals of Increase and Decrease

Understanding intervals of increase and decrease of a function is essential for analyzing a function's behavior. These intervals indicate where the function is moving upwards or downwards as the input values, or 'x' values, increase.

To determine these intervals for the function given by the exercise, the graph is observed to identify where the function is rising or falling. For the function f(x) = x \(3-x\), to find the intervals of increasing or decreasing, one would look at the slope of the function. If the slope is upwards as 'x' increases, the function is increasing; if the slope is downwards, then the function is decreasing.

In this function, since there is no interval where the function's slope is zero or negative, the only interval of increase is from 0 to 3. There is no decrease or constant behavior noted in the given domain. Thus, you can describe the behavior of the function in the interval [0, 3] as increasing throughout.

Analyzing Function Behavior

Visual Inspection and Calculus Methods

To further analyze the behavior of a function beyond just identifying intervals of increase and decrease, one might study the graph's concavity, look for asymptotes, and recognize any critical points where the function's derivative equals zero or does not exist. With the function f(x) = x \(3 - x\), visual inspection of the graph reveals the function's general shape and behavior within its domain.

Calculus tools, such as derivatives, can be used for a more detailed analysis. The derivative of a function can tell us where the function's slope is positive (increasing), negative (decreasing), or zero (constant or at a critical point). Analyzing the derivative helps predict the function's behavior more accurately than by observation alone.

Domain of a Function

The domain of a function comprises all the input values, or 'x' values, for which the function is defined. In essence, it tells us the set of all possible 'x' values that we can plug into the function without causing any contradictions or undefined expressions.

In the provided exercise, f(x) = x \(3 - x\), there are restrictions due to the square root operation. Square roots require their input to be non-negative; hence, for \(3 - x\), 'x' must be less than or equal to 3. At the same time, because 'x' is also outside the radical, 'x' must be greater than or equal to 0. As a result, the domain of this function is limited to the closed interval [0, 3] where both conditions are satisfied. This information is crucial when graphing the function, as it limits the range of 'x' values we consider.