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Chapter 2: Problem 28

Make a table listing ordered pairs that satisfy each equation. Then graph the equation. Determine the domain and range, and whether \(y\) is a function of \(x .\) $$y=-x^{3}$$

Short Answer

Expert verified
The equation is a function with domain and range both \(\mathbb{R}\).

Step by step solution

01

- Understanding the equation

First, recognize that the given equation is a cubic function: \(y = -x^{3}\). In this function, for each value of \(x\), there is a corresponding value of \(y\).
02

- Create a table of ordered pairs

Select different values for \(x\) and compute the corresponding \(y\) values using the equation. For example, you can choose \(x = -2, -1, 0, 1, 2\): | \(x\) | \(y = -x^{3}\) ||------|---------------|| -2 | -(-2)^{3} = 8 || -1 | -(-1)^{3} = 1 || 0 | -(0)^{3} = 0 || 1 | -(1)^{3} = -1 || 2 | -(2)^{3} = -8 |
03

- Plot the points on a graph

Using the table from Step 2, plot the pairs (-2, 8), (-1, 1), (0, 0), (1, -1), and (2, -8) on the coordinate plane.
04

- Draw the curve

After plotting the points, draw a smooth curve through them to represent the equation \(y = -x^{3}\).
05

- Determine the domain and range

For the function \(y = -x^{3}\), the domain (all possible \(x\) values) is all real numbers, denoted as \(\mathbb{R}\). The range (all possible \(y\) values) is also all real numbers, \(\mathbb{R}\).
06

- Check if y is a function of x

For each \(x\) value, there is exactly one \(y\) value. Therefore, \(y\) is a function of \(x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordered Pairs
To understand ordered pairs, remember that each pair consists of two elements: \(x\) and \(y\). For every \(x\) value you choose in the equation \(y = -x^{3}\), there is a corresponding \(y\) value. This relationship is written as \( (x, y) \).

For instance, by choosing \(x = -2\), you get \(y = 8,\) making the ordered pair \( (-2, 8) \). Repeating this process for \(x = -1, 0, 1, \text{ and}\ 2\) gives ordered pairs such as \( (-1, 1), (0, 0), (1, -1),\) and \( (2, -8) \).

In summary, an ordered pair shows how two numbers relate in a function, where the first number is from the domain and the second from the range.
Graphing Functions
Graphing makes it easier to understand the shape and behavior of a function. For the cubic function \(y = -x^{3}\), the process begins with plotting the ordered pairs found in the previous step.

Follow these tips for graphing:
  • Locate and mark each \( (x, y) \) point on the coordinate plane. For instance, plot \( (-2, 8) \) by moving \ -2 \ units on the x-axis and \ 8 \ units on the y-axis.
  • Continue plotting all pairs such as \( (-1, 1), (0, 0), (1, -1), \text{ and}\ (2, -8) \).
  • Once all points are plotted, draw a smooth curve passing through them. This curve represents the cubic function accurately.
Understanding the graph helps visualize how \(y\) changes as \(x\) varies.
Domain and Range
The domain and range of a function define the set of all possible input and output values, respectively.

For the cubic function \(y = -x^{3}\):
  • Domain: Since any real number can be cubed, the domain is all real numbers, denoted as \( \text{R} \).
  • Range: Likewise, since a cubic function can output any real number, the range is also \( \text{R} \).
This means you can plug in any value for \(x\), and it will produce a corresponding \(y\) value in this function.
Function Analysis
Analyzing the function \(y = -x^{3}\) provides a deeper insight into its properties and behavior.

One key aspect is understanding if \(y\) is a function of \(x\). Simply put, for every value of \(x\) in a function, there should be exactly one corresponding \(y\) value. In the case of \(y = -x^{3}\), this relationship holds true because:
  • For every \(x\) value, computing \(-x^{3}\) gives a unique \(y\) value.
  • This indicates a clear one-to-one correspondence.
Therefore, \(y\) is indeed a function of \(x\).

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