Problem 97 - Elementary and Intermediate Algebra

Question

Complete each table. See Example 11. $$ \begin{array}{|c|c|} \hline x & {x^{3}-1} \\ \hline 0 & {} \\ \hline-1 & {} \\ \hline-3 & {} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

For \( x = 0 \), \( x^3 - 1 = -1 \); for \( x = -1 \), \( x^3 - 1 = -2 \); for \( x = -3 \), \( x^3 - 1 = -28 \).

Step 1: Understanding the Function

We need to calculate \( x^3 - 1 \) for each value given in the table. The function provided is \( f(x) = x^3 - 1 \). We’ll substitute each \( x \) value into this function to find the corresponding output.

Step 2: Calculate for \( x = 0 \)

Substitute \( x = 0 \) into the function: \( f(0) = 0^3 - 1 \). This simplifies to \( 0 - 1 = -1 \). Thus, when \( x = 0 \), \( f(0) = -1 \).

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

Substitute \( x = -1 \) into the function: \( f(-1) = (-1)^3 - 1 \). This simplifies to \( -1 - 1 = -2 \). Thus, when \( x = -1 \), \( f(-1) = -2 \).

Step 4: Calculate for \( x = -3 \)

Substitute \( x = -3 \) into the function: \( f(-3) = (-3)^3 - 1 \). This simplifies to \( -27 - 1 = -28 \). Thus, when \( x = -3 \), \( f(-3) = -28 \).

Step 5: Complete the Table

Based on the calculations we've done: - For \( x = 0 \), \( f(0) = -1 \) - For \( x = -1 \), \( f(-1) = -2 \) - For \( x = -3 \), \( f(-3) = -28 \). Fill these values in the table accordingly.

Key Concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. When we see a number raised to the power of three, as in the cases of the given function, this means multiplying the number three times. For instance:

• \( x^3 \) means \( x \times x \times x \).

In the exercise, each value of \( x \) is used to compute \( x^3 \). Let's look at some examples:

• For \( x = 0 \), \( 0^3 = 0 \).
• For \( x = -1 \), \( (-1)^3 = -1 \) because multiplying three negative numbers results in a negative number.
• For \( x = -3 \), \( (-3)^3 = -27 \) due to multiplying three negative threes together.

Understanding how exponents work, especially with negative numbers, is essential for function evaluation.

Function Evaluation

Function evaluation is the process of finding the output value of a function for a given input value. In this exercise, we evaluate the function \( f(x) = x^3 - 1 \). To evaluate this function:

• First, substitute the given \( x \) value into the function.
• Then, carry out the calculation using the rules of arithmetic, such as exponentiation and subtraction.

Let's apply this to the exercise:

• For \( x = 0 \), substituting gives us \( 0^3 - 1 = -1 \).
• For \( x = -1 \), substituting gives us \( (-1)^3 - 1 = -2 \).
• For \( x = -3 \), substituting gives us \( (-3)^3 - 1 = -28 \).

By evaluating the function at each \( x \) value, we obtain the results that are used to complete the table.

Negative Numbers

Negative numbers can sometimes be tricky, especially when combined with exponents and other operations. Understanding negative numbers in this context helps us ensure our calculations are accurate.
When multiplying or raising negative numbers to an odd power, the result remains negative. Here's why:

• Take \( (-1)^3 \): The calculation is \( -1 \times -1 \times -1 = -1 \).
• For \( (-3)^3 \): The calculation is \( -3 \times -3 \times -3 = -27 \). Each negative triple multiplication results in a negative outcome.

Adding or subtracting with negative numbers follows straightforward rules:

• Subtracting 1 from any negative number makes it more negative. For \( -2 - 1 = -3 \), removing another positive amount moves further down the negative scale.

These operations, while they seem simple, are easy to get wrong if not considered carefully. Mastering these concepts is pivotal when dealing with algebraic functions.