Problem 110 - College Algebra Essentials

Question

use a graphing utility to graph each function. Use \(a[-5,5,1]\) by \([-5,5,1]\) viewing rectangle. Then find the intervals on which the function is increasing, decreasing, or constant. $$ f(x)=x^{\frac{1}{3}}(x-4) $$

Step-by-Step Solution

Answer

The function \(f(x) = x^{\frac{1}{3}}(x-4)\) is increasing in the intervals \(-∞ < x ≤ 0\) and \(4 ≤ x < ∞\), and decreasing in the interval \(0 < x ≤ 4\). The function is never constant.

Step 1: Graph the Function

Start by entering the function \(f(x) = x^{\frac{1}{3}}(x-4)\) into the graphing tool of your choice. Ensure the viewing window is set to a[-5,5,1] by [-5,5,1] to correctly display the graph.

Step 2: Identify the Intervals

Observing the graph, we look for when the function is increasing and decreasing. We can say a function is increasing when the slope of the line is positive, decreasing when the slope is negative and constant when the slope is zero.

Step 3: Analyze for Increasing Intervals

The graph of the function is increasing from negative infinity to 0 and from 4 to positive infinity as we move from left to right. Hence, the function is increasing on the interval \(-∞ < x ≤ 0\) and \(4 ≤ x < ∞\).

Step 4: Analyze for Decreasing Intervals

The graph of the function decreases from 0 to 4 as we move from left to right. Hence, the function is decreasing on the interval \(0 < x ≤ 4\).

Step 5: Analyze for Constant Intervals

The graph of the function does not stay constant at any interval.

Key Concepts

Analyzing Intervals

To effectively analyze intervals in a function's graph, one must understand the behavior of the function within specific ranges.

When investigating the graph of a function like the function given in the exercise, which is \(f(x) = x^{\frac{1}{3}}(x-4)\), you're looking for sections along the x-axis where the function's output (y-value) either increases, decreases, or remains constant as 'x' increases. To do this, you must observe the slope of the function graph. A positive slope denotes an increasing function, while a negative slope points to a decreasing function. An area of the graph where the line is flat (horizontal) indicates that the function is constant on that interval.

In the context of the given exercise, by following these guidelines and observing the graph, you can determine that there are no constant intervals and identify the specific intervals of increase and decrease, which are crucial for understanding the function's overall behavior.

Increasing and Decreasing Functions

The terms 'increasing' and 'decreasing' help to describe the direction a function's values are heading as you read the graph from left to right. An increasing function means that as