Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function and use the Horizontal Line Test to determine whether the function is one-to-one and so has an inverse function. $$h(x)=|x|-|x-4|$$

Step-by-Step Solution

Answer

After graphing the function and implementing the Horizontal Line Test, the conclusion about whether \( h(x) = |x| - |x-4| \) has an inverse function or not can be drawn. This will be confirmed by the number of times a horizontal line intersects with the graph of the function.

Step 1: Graph the Function

Start by graphing the function \( h(x) = |x| - |x-4| \) using a graphing utility. Remember, the absolute value function will create a 'V' shape on the graph where the point of the 'V' is at the x-value that makes the inside of the absolute value zero.

Step 2: Implement the Horizontal Line Test (HLT)

Draw horizontal lines on the graph. If any line intersects the graph at more than one point, the function is not one-to-one.

Step 3: Evaluate test results

Analyze the graph after applying the HLT. If the horizontal line intersects the graph only once, it can confirm that the function is one-to-one and, thus, has an inverse function.

Key Concepts

Graphing Utilities

Graphing utilities are incredibly helpful tools for visualizing mathematical functions, making this process much easier and more interactive. When you use a graphing calculator or software, you can input a function and see a detailed graph almost instantly. This visual representation helps you understand the function's behavior at various intervals.

• Graphing utilities allow you to zoom in and out, displaying the function on different scales to observe fine details or broader patterns.
• You can plot multiple functions simultaneously to compare their shapes and intersections.
• Dynamic graphing utilities may also allow for parameter changes to see real-time effects on the graph.

For the function \( h(x) = |x| - |x-4| \), graphing utilities graph it as a 'V' shape. By entering this function, the tool displays the graph where points of interest, such as where the graph changes direction or crosses the axes, become evident. Such visualization aids in comprehending the underlying concepts of the function and assists in tasks like the Horizontal Line Test, which evaluates whether a function is one-to-one.

Horizontal Line Test

The Horizontal Line Test is a graphical method for determining whether a function is one-to-one, meaning it has a unique inverse. A function is one-to-one if and only if no horizontal line intersects its graph at more than one point.

• To perform the test, draw several horizontal lines across the function's graph.
• If any line touches the graph more than once, the function fails the test and is not one-to-one.
• A function that passes this test can have an inverse function, as each output value is linked to exactly one input value.

For the function \( h(x) = |x| - |x-4| \), plot horizontal lines across the 'V' shaped graph. If you observe that all horizontal lines intersect the graph at most once, then the graph is one-to-one and the function can indeed have an inverse. However, if some lines cross the graph more than once, the function does not have an inverse. This test is crucial for confirming the function’s status as one-to-one and confirming its invertibility.

Absolute Value Functions

Absolute value functions are fundamental and often appear in various mathematical contexts. An absolute value \(|x|\) indicates the distance of a number from zero on the number line, always yielding a non-negative result, which is why their graphs typically depict a 'V' shape.

• The vertex of the 'V' occurs at the point where the expression inside the absolute value equals zero.
• In the case of \( |x| \), the vertex is at \((0, 0)\), while for \( |x-4| \), it shifts to \((4, 0)\).
• These functions often reflect over the x-axis, as the absolute value function counts both negative and positive inputs equally due to its non-negative result.
• The piecewise nature of absolute value functions means they can be understood better by splitting them into different linear functions over different intervals.

For \(h(x) = |x| - |x-4|\), the function combines two absolute value expressions, creating complex but symmetrical graphs about specific points. Understanding how these combine helps in solving such functions and using tests like the Horizontal Line Test to determine properties like invertibility. Graphing utilities effectively showcase these properties, providing clear imagery to aid learning.