Problem 20 - College Algebra with Corequisite Support

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

Yes, the function \( f(x) = \sqrt[3]{3x + 1} \) is one-to-one as it passes the horizontal line test.

Step 1: Understanding One-to-One Functions

A function is one-to-one if every output value is produced by exactly one input value. In other words, different inputs should produce different outputs. A graphical test for this property is the horizontal line test: a function is one-to-one if no horizontal line intersects the graph more than once.

Step 2: Graphing the Function

Plot the function \( f(x) = \sqrt[3]{3x + 1} \) using a graphing utility. Ensure that you use a wide enough domain to capture the function's behavior, typically covering both positive and negative values of \( x \).

Step 3: Applying the Horizontal Line Test

Visually inspect the plotted function. Check if any horizontal line drawn on the graph intersects the curve more than once. If every horizontal line intersects the graph at most once, the function is one-to-one.

Step 4: Conclusion

After performing the horizontal line test, observe that no horizontal line cuts the graph of \( f(x) = \sqrt[3]{3x + 1} \) more than once. Hence, the function passes the test for being one-to-one.

Key Concepts

Horizontal Line Test

The horizontal line test is an efficient graphical method used to determine if a function is one-to-one. A function is deemed one-to-one if every horizontal line crosses the graph of the function at most once. This simple visual test can be quite revealing. If any horizontal line crosses the function's graph more than once, the function fails the test and is not one-to-one.
This test is particularly useful because:

• It allows for a quick visual inspection.
• Requires no complex calculations.
• Encourages understanding of function symmetry and behavior.

When you apply the horizontal line test to a graph, imagine drawing infinite horizontal lines across different sections of the graph.
If none intersect the curve more than once, the function successfully passes the test, confirming its one-to-one nature.

Graphical Test

Using a graphical test to evaluate a function’s properties is a practical approach, especially in conjunction with modern graphing utilities.
A graphical test involves plotting the function on a coordinate plane and visually inspecting it for specific features or properties. Graphing utilities help make this process more precise and accessible:

• You can adjust the zoom and scale to observe the behavior over a wide range of values.
• Easy navigation of complex function domains and visual exploration of function behavior.
• Immediate feedback and results, allowing for quick verification of hypotheses or solutions.

These tools illuminate a function’s characteristics, such as increasing or decreasing intervals, symmetry, and continuity, which are vital in understanding its one-to-one nature.

Function Graphing

Function graphing is a critical skill in mathematics, aiding in the visual representation of equations or functions. To graph a function like \( f(x) = \sqrt[3]{3x + 1} \), you follow certain steps:

• Select a range of \( x \) values (covering both positive and negative) to grasp the function’s behavior.
• Calculate corresponding \( y \) values using the function’s formula.
• Plot these \( (x, y) \) points on a coordinate graph.
• Connect the points smoothly to represent the function’s continuous nature.

Graphing not only helps in applying tests like the horizontal line test but also deepens understanding of function features, such as turning points, asymptotes, and domain restrictions.
This visual approach makes abstract mathematical concepts much more accessible and understandable.