Problem 84 - Elementary and Intermediate Algebra

Question

$$ \begin{aligned} &f(x)=-\sqrt[3]{x}\\\ &\begin{array}{|r|r|} \hline x & y \\ \hline-8 & \\ -1 & \\ 0 & \\ 1 & \\ 8 & \\ \hline \end{array} \end{aligned} $$

Step-by-Step Solution

Answer

The completed values in the table are: \((-8, 2)\), \((-1, 1)\), \((0, 0)\), \((1, -1)\), and \((8, -2)\).

Step 1: Understand the Function

The given function is \( f(x) = -\sqrt[3]{x} \). This means for each \( x \) value, you need to find the cube root of \( x \) and then multiply it by -1.

Step 2: Calculate the Function for \( x = -8 \)

Calculate \( f(-8) \). The cube root of \(-8\) is \(-2\), since \( (-2)^3 = -8 \). Therefore, \( f(-8) = -(-2) = 2 \).

Step 3: Calculate the Function for \( x = -1 \)

Calculate \( f(-1) \). The cube root of \(-1\) is \(-1\), since \( (-1)^3 = -1 \). Therefore, \( f(-1) = -(-1) = 1 \).

Step 4: Calculate the Function for \( x = 0 \)

Calculate \( f(0) \). The cube root of \( 0 \) is \( 0 \), since \( 0^3 = 0 \). Therefore, \( f(0) = -(0) = 0 \).

Step 5: Calculate the Function for \( x = 1 \)

Calculate \( f(1) \). The cube root of \( 1 \) is \( 1 \), since \( 1^3 = 1 \). Therefore, \( f(1) = -(1) = -1 \).

Step 6: Calculate the Function for \( x = 8 \)

Calculate \( f(8) \). The cube root of \( 8 \) is \( 2 \), since \( 2^3 = 8 \). Therefore, \( f(8) = -(2) = -2 \).

Step 7: Fill in the Table

Use the results from Steps 2 to 6 to fill in the table:\[\begin{array}{|r|r|}\hlinex & y \hline-8 & 2 \-1 & 1 \0 & 0 \1 & -1 \8 & -2 \\hline\end{array}\]

Key Concepts

Function Evaluation

Cube root functions may look complicated at first glance, but they're simpler than they seem. Evaluating a function means finding what the function's formula gives us for particular input values. For our function, we use the formula: \[ f(x) = -\sqrt[3]{x} \]This means we take the cube root of any given number \(x\) and then apply a negative sign to the result. To evaluate the function, consider the steps:

• Find the cube root of \(x\). The cube root \(\sqrt[3]{x}\) is a value that, when raised to the power of 3, gives \(x\) back.
• Apply the negative sign to the cube root. The result is the final value of the function.

For each step, we need to carefully evaluate the cube root first and remember that the outside negative sign changes its direction. Function evaluation is a way to transform inputs \(x\) into results \(y = f(x)\).

Negative Sign Application

The negative sign in our function formula is crucial and actually determines the nature of the outputs. Understanding this involves recognizing two steps:- First, we find the cube root of the given \(x\) value. - Then, we multiply this cube root by \(-1\), flipping its value from positive to negative or vice versa.For instance, if our output from the cube root function is positive, say \(2\) for \(x = 8\), it becomes \(-2\) after applying the negative sign from the original function. Similarly, if \(x\) is negative, like \(-8\), where the cube root is \(-2\), it turns into a positive value upon the negative sign application, resulting in \(2\).This step is easy to overlook, but recognizing it allows you to accurately determine the final output for each value of \(x\).

Table of Values

A table of values is an organized way of presenting function evaluations for different inputs. It is not only helpful for visual learners but also crucial for verifying our calculations. For the function \(f(x) = -\sqrt[3]{x}\), we can input our evaluated values into the table format:

• For \(x = -8\), we calculated \(y = 2\).
• For \(x = -1\), we found \(y = 1\).
• For \(x = 0\), the result was \(y = 0\).
• For \(x = 1\), we have \(y = -1\).
• Finally, for \(x = 8\), the output is \(y = -2\).

Each of these steps involves checking our function evaluation skills and understanding the application of the negative sign, helping us predict behaviors of the function further beyond specific points. By visualizing these pairs in a tabular form, it provides a clear summary and makes it easier to identify patterns or irregularities in function behavior.