Problem 32 - Elementary and Intermediate Algebra

Question

Graph each function by creating a table of function values and plotting points. Give the domain and range of the function. See Examples \(2,3,\) and 4. $$ f(x)=x^{3}+2 $$

Step-by-Step Solution

Answer

Domain: \((-\infty, \infty)\), Range: \((-\infty, \infty)\).

Step 1: Create a Table of Values

To graph the function \( f(x) = x^3 + 2 \), start by creating a table of values. Choose a range of \( x \)-values, typically including positive and negative ones, plus zero. Here is a suggested set of \( x \)-values: \(-2, -1, 0, 1, 2\).

Step 2: Calculate \( f(x) \) Values

Use the function \( f(x) = x^3 + 2 \) to calculate \( f(x) \) for each \( x \) value.- For \( x = -2 \), \( f(-2) = (-2)^3 + 2 = -8 + 2 = -6 \)- For \( x = -1 \), \( f(-1) = (-1)^3 + 2 = -1 + 2 = 1 \)- For \( x = 0 \), \( f(0) = 0^3 + 2 = 2 \)- For \( x = 1 \), \( f(1) = 1^3 + 2 = 1 + 2 = 3 \)- For \( x = 2 \), \( f(2) = 2^3 + 2 = 8 + 2 = 10 \)

Step 3: Plot Points on a Graph

Now, plot the points from the table on a coordinate plane: - \((-2, -6)\)- \((-1, 1)\)- \((0, 2)\)- \((1, 3)\)- \((2, 10)\)Draw a smooth curve through the points, which reflects the cubic nature of the function.

Step 4: Determine the Domain

The domain of the function \( f(x) = x^3 + 2 \) includes all real numbers because there are no restrictions on the values \( x \) can take for a cubic function. Therefore, the domain is: \( \text{Domain: } (-\infty, \infty) \)

Step 5: Determine the Range

Since the cubic function \( x^3 \) stretches from negative infinity to positive infinity, and adding 2 shifts this graph up without affecting its range, the range of \( f(x) = x^3 + 2 \) is also all real numbers. Therefore, the range is: \( \text{Range: } (-\infty, \infty) \)

Key Concepts

Domain and Range

The domain and range are essential concepts when working with functions. The domain of a function refers to the set of all possible input values (typically denoted as \( x \)-values) for which the function is defined. For cubic functions like \( f(x) = x^3 + 2 \), the domain is all real numbers. This means that you can plug any real number into the function and obtain a valid output.
The reason cubic functions have a domain of all real numbers is because there's no operation or restriction in the function \( f(x) = x^3 + 2 \) that would disqualify any real number. No division by zero or square roots of negative numbers are involved, ensuring a smooth continuity of input values.
The range of a function is the set of all possible outputs, or \( f(x) \)-values, after inputting the domain values into the function. For cubic functions like ours, the range is also all real numbers, \((-\infty, \infty)\). As \( x \) approaches positive or negative infinity, the function \( f(x) = x^3 + 2 \) also approaches positive or negative infinity. Therefore, the function can achieve any real value.

Plotting Graphs

Plotting the graph of a function helps in visualizing how the function behaves for various \( x \)-values. To plot the graph of the cubic function \( f(x) = x^3 + 2 \), we need to start by plotting points.

• Identify suitable \( x \)-values, including both negative and positive values, and zero. A good spread could be \(-2, -1, 0, 1, 2\).
• Calculate the corresponding \( f(x) \)-values for each \( x \), for example:
  • For \( x = -2 \), calculate: \( f(-2) = -6 \)
  • For \( x = -1 \), calculate: \( f(-1) = 1 \)
  • And so forth.
• Once the values are computed, plot each point \((x, f(x))\) on a coordinate graph.

Connect these plotted points smoothly. Since \( f(x) = x^3 + 2 \) is a cubic function, the plot has a characteristic shape, smoothly curving and stretching from quadrants III to I in the coordinate plane. Cubic function graphs are represented by a curve that steadily increases or decreases based on the \( x \)-inputs.

Function Tables

Using function tables is an excellent first step in graphing functions like cubic ones. A function table helps you organize \( x \)-values and their corresponding \( f(x) \)-values neatly, making the plotting process more straightforward.
To create a table for \( f(x) = x^3 + 2 \), follow these steps:

• Select a range of \( x \)-values to input into the function. You often choose a mix of negative, zero, and positive numbers, such as \(-2, -1, 0, 1, 2\).
• For each \( x \)-value, compute \( f(x) \). For instance:
  • \( x = -2 \rightarrow f(-2) = -6 \)
  • \( x = 0 \rightarrow f(0) = 2 \)
  • Continue this for each selected point.

This comprehensive display offers a clear view of how \( x \)-values relate to \( f(x) \)-values, which serves as a blueprint for making accurate graphs. It simplifies drawing the curve and understanding the function's behavior.