Problem 20 - College Algebra

Question

For the following exercises, use a graphing utility to determine whether each function is one-to-one. $$ f(x)=\sqrt[3]{3 x+1} $$

Step-by-Step Solution

Answer

The function \( f(x) = \sqrt[3]{3x + 1} \) is one-to-one.

Step 1: Understand the concept of a one-to-one function

A function is one-to-one if every y-value in the function’s range corresponds to exactly one x-value in its domain. Graphically, this means that the graph of the function would pass the horizontal line test, indicating that no horizontal line intersects the graph more than once.

Step 2: Analyze the function

The given function is \( f(x) = \sqrt[3]{3x + 1} \). This is a cube root function, which are typically one-to-one because they increase or decrease steadily along their domains.

Step 3: Use a graphing utility

Use a graphing calculator or an online tool to plot the graph of \( f(x) = \sqrt[3]{3x + 1} \). Observe the behavior of the graph to see if it passes the horizontal line test.

Step 4: Apply the horizontal line test

Check visually or logically if any horizontal line crosses the graph of \( f(x) = \sqrt[3]{3x + 1} \) more than once. Since this is a cube root function, it will pass the test because it is monotonic, either strictly increasing or strictly decreasing.

Key Concepts

Horizontal Line Test

The horizontal line test is a simple way to determine if a function is one-to-one. If a function is one-to-one, it means that each output value (y-value) corresponds to exactly one input value (x-value). This is visually represented in the graph of the function.

To perform this test, draw horizontal lines across different parts of the graph. If at any point, a horizontal line intersects the graph in more than one place, the function is not one-to-one. For one-to-one functions, such horizontal lines will only touch the graph at one point each.

In the case of the function \( f(x) = \sqrt[3]{3x + 1} \), we can apply this test by observing that it is a cube root function. Cube root functions typically change steadily and are either entirely increasing or decreasing across their domain.

Thus, when you draw horizontal lines across its graph, they will touch it only once. This means that the function passes the horizontal line test, confirming it is one-to-one.

Cube Root Functions

Cube root functions are a type of function that involve the cube root of an expression, often written as \( f(x)=\sqrt[3]{ax+b} \). These functions have distinct properties, which make them fairly easy to recognize and analyze.

Key characteristics of cube root functions include:

• They are defined for all real numbers. This means you can plug any value of \( x \) into the function, and you will get a valid output.
• They are continuous and smooth, meaning there are no gaps, jumps, or sharp corners in their graphs.
• They typically exhibit a monotonic behavior. This means they either increase or decrease steadily, which is why they usually are one-to-one functions.

For \( f(x) = \sqrt[3]{3x + 1} \), the function is monotonic and steadily increasing. This guarantees that any horizontal line will intersect the graph only once, satisfying the one-to-one condition.

Graphing Utilities

Graphing utilities are powerful tools that help visualize the behavior of functions. These utilities can range from handheld graphing calculators to sophisticated software applications or online graphing tools.

Benefits of using graphing utilities include:

• Quick visualization: Instantly see the shape of the graph, helping you understand the function's behavior, including whether it is increasing or decreasing.
• Accuracy: Precision plotting can help you check theoretical predictions against the visual graph.
• Interactive features: Some utilities allow zooming or tracing, making it easier to explore specific parts of the graph.

For the function \( f(x) = \sqrt[3]{3x + 1} \), using a graphing utility can quickly show its monotonic nature and confirm whether it is one-to-one by passing the horizontal line test. This makes graphing utilities invaluable for both analyzing functions and in complex mathematical investigations.