Problem 36 - 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. $$ f(x)=(x-1)^{3} $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Function and Its Characteristics

We are given the function \( f(x) = (x-1)^3 \). This is a cubic function where the input \( x \) is shifted to the right by 1 unit, due to \( x-1 \), and is then cubed. The general shape of the graph will be an S-shaped curve due to its cubic nature.

Step 2: Create a Table of Function Values

Select a range of \( x \) values around the point \( x = 1 \) (say from \( x = -1 \) to \( x = 3 \)) and compute the corresponding \( f(x) \) values.- \( x = -1 \), \( f(-1) = ((-1)-1)^3 = -8 \) - \( x = 0 \), \( f(0) = (0-1)^3 = -1 \) - \( x = 1 \), \( f(1) = (1-1)^3 = 0 \) - \( x = 2 \), \( f(2) = (2-1)^3 = 1 \) - \( x = 3 \), \( f(3) = (3-1)^3 = 8 \)

Step 3: Plot the Points

Using the table from step 2, plot the points \((-1, -8)\), \((0, -1)\), \((1, 0)\), \((2, 1)\), and \((3, 8)\) on a graph. You will see the characteristic S-shaped curve, which centers around the point \( (1, 0) \).

Step 4: Determine the Domain of the Function

The domain of \( f(x) = (x-1)^3 \) is all real numbers \( (-\infty, \infty) \), since cubic functions are defined for all real \( x \).

Step 5: Determine the Range of the Function

The range of \( f(x) = (x-1)^3 \) is also all real numbers \( (-\infty, \infty) \), because as \( x \) takes all real values, \( (x-1)^3 \) can also take any real value.

Key Concepts

Domain and Range

When discussing graphs of functions, the **domain** and **range** are essential concepts to understand. The **domain** refers to all the possible input values (or x-values) that a function can accept. For a cubic function like \( f(x) = (x-1)^3 \), the domain is straightforward. Cubic functions are defined for all real numbers, meaning that this function can take any real number as its input. In mathematical terms, we express this as the domain being \((-infty, infty)\).

On the other hand, the **range** represents all possible output values (or f(x) values) that the function can produce. Just like the domain, the range of a cubic function like \( f(x) = (x-1)^3 \) is also all real numbers. As \( x \) varies over all real numbers, the expression \((x-1)^3 \) can output every possible real number. Thus, the range is also \((-infty, infty)\).

Understanding these concepts is vital as they determine the inputs we use and the outputs we can expect in any real-world application of cubic functions.

Function Table

Constructing a **function table** is a helpful way to organize input and output values when graphing functions. Let’s break down how to create a function table for the cubic function \( f(x) = (x-1)^3 \).

1. **Choose Your X-Values**: It’s beneficial to select x-values around the function's critical points. For the function \( f(x) = (x-1)^3 \), x-values around 1, such as -1, 0, 1, 2, and 3, give a clear picture of the graph's behavior.
2. **Calculate Corresponding f(x) Values**: Substitute each x-value into the function to find the corresponding f(x) values. For instance:
- When \( x = -1 \), \( f(x) = (-1-1)^3 = -8 \)
- When \( x = 0 \), \( f(x) = (0-1)^3 = -1 \)
- When \( x = 1 \), \( f(x) = (1-1)^3 = 0 \)
- When \( x = 2 \), \( f(x) = (2-1)^3 = 1 \)
- When \( x = 3 \), \( f(x) = (3-1)^3 = 8 \)
3. **Organize and Plot**: Write these pairs down in a table and plot them on a graph. This step allows you to visualize the function's pattern and gain a deeper understanding of its shape.

S-Shaped Curve

A **cubic function** like \( f(x) = (x-1)^3 \) is known for its distinctive **S-shaped curve** or "sigmoid curve." Understanding this shape is crucial for graphing.

The S-shape occurs because:

• **Inflection Point** – The graph transitions from curving downwards to upwards at its inflection point. For \( f(x) = (x-1)^3 \), this is at the point \((1, 0)\).


• **Symmetrical Around the Inflection Point** – The curve is typically symmetric around this point, illustrating balanced growth and decay.
  
• **Gradual Change** – The slope of the curve changes gradually. As \( x \) moves away from its inflection point, the increase or decrease slows down.
  

This shape is significant because it reflects many natural growth patterns, making the cubic graph applicable in fields such as biology and economics. In these disciplines, recognizing an S-shaped curve can help model complex systems effectively. Demonstrating proficiency in identifying and drawing these curves enriches the study of functions, emphasizing their real-world implications.