Problem 29 - A Graphical Approach to College Algebra

Question

Graph each function in the standard viewing window of your calculator, and trace from left to right along a representative portion of the specified interval. Then fill in the blank of the following sentence with either increasing or decreasing. Over the interval specified, this function is __________. $$f(x)=-\sqrt[3]{x} ;(-\infty, \infty)$$

Step-by-Step Solution

Answer

decreasing

Step 1: Understand the function

The function is given as \( f(x) = -\sqrt[3]{x} \). This means the function involves taking the cube root of \( x \) and then changing its sign.

Step 2: Evaluate the sign of the function

Since this is a cube root function, the cube root of a negative number is negative and the cube root of a positive number is positive. However, due to the negative sign in front, the function \( f(x) = -\sqrt[3]{x} \) will return a negative value for positive \( x \) and a positive value for negative \( x \).

Step 3: Analyze the behavior over the interval

The interval in question is \( (-\infty, \infty) \). For cube root functions, as \( x \) increases, \( \sqrt[3]{x} \) also increases. However, since the function has a negative sign, as \( x \) increases, \( f(x) \) decreases.

Step 4: Confirm the decreasing behavior

By plotting the graph of \( f(x) \) on a calculator, starting from a large negative value and moving towards a large positive value, we can trace the function and observe that it constantly moves downward, confirming it is decreasing.

Key Concepts

Cube Root Function

Understanding the cube root function is important as it's one of the foundational concepts in mathematics. A cube root function is an inverse operation of cubing a number. It finds the value that, when multiplied by itself three times, gives the original number.

• For example, the cube root of 8 is 2, because \[2 \times 2 \times 2 = 8\].
• The cube root of -8 is -2 because \(-2 \times -2 \times -2 = -8\).

In mathematical terms, the cube root of \(x\) is written as \(\sqrt[3]{x}\). When analyzing the function \(f(x) = -\sqrt[3]{x}\), it's essential to note that the negative sign in front reflects the graph across the x-axis, inverse to what you might initially expect. This flipping affects the way the graph appears, and subsequently how the function behaves over its domain.

Interval Analysis

Interval analysis involves understanding how a function behaves over a specified range of x-values. For a function like \(f(x) = -\sqrt[3]{x}\), every x-value within the interval \((-\infty, \infty)\) has a corresponding y-value. Cube root functions are unique because they are defined for all real numbers. They smoothly pass through the origin, and for \(f(x) = -\sqrt[3]{x}\), the function will consistently give outputs along its path:

• As \(x\) progresses from a negative infinity to zero, \(f(x)\) rises since it's the reflection.
• When continuing from zero to positive infinity, \(f(x)\) starts decreasing due to the negative sign.

This function exemplifies the idea of a function changing behavior based on a negative sign, flipping its general cube root behavior. For \(f(x) = -\sqrt[3]{x}\), the entire interval shows a decreasing trend, moving downwards on the graph as \(x\) increases.

Graphical Calculator Usage

Using a graphical calculator can greatly simplify the process of understanding complex functions. With a calculator set to a standard viewing window, you can easily track how a function behaves over a given interval.

• First, enter the function \(f(x) = -\sqrt[3]{x}\) into the calculator.
• Next, observe the function's graph as it displays. Follow the path of the function from left to right, noting the trajectory.

The goal is to verify your theoretical analysis—such as the decreasing nature of this particular function—by visually confirming it. With a calculator, watch as the function begins high and trends lower as \(x\) increases, perfectly illustrating the concept of decreasing function behavior over \((-\infty, \infty)\). Graphical calculators are powerful tools for visual learners, providing an immediate connection to the algebraic expressions involved.