Problem 49 - 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. $$h(x)=\sqrt{16-x^{2}}$$

Step-by-Step Solution

Answer

The function \(h(x) = \sqrt{16-x^2} \) is one-to-one and has an inverse. This conclusion is supported by graphing the function and seeing that it passes the Horizontal Line Test.

Step 1: Graph the function

First, plot the function \(h(x) = \sqrt{16-x^2} \) using a graphing utility.

Step 2: Perform the Horizontal Line Test

Draw horizontal lines through the graph of the function. If any horizontal line intersects the graph at more than one point, the test fails, and the function is not one-to-one.

Step 3: Determine whether the function is one-to-one

If the function passes the Horizontal Line Test, it is one-to-one, meaning each input corresponds to exactly one output, and the function has an inverse.

Key Concepts

Graphing Utility

Graphing utilities are powerful tools that help visualize mathematical functions. They can range from simple online calculators to advanced graphing calculators or software programs like Desmos, GeoGebra, or similar apps. These tools allow you to input a mathematical function and instantly see its graph. For the function given, which is \( h(x) = \sqrt{16-x^2} \), when entered into a graphing utility, you'll see a semicircle that represents this function.

• Graphs quickly and accurately represent the function.
• Visual aids can make it easier to understand and analyze behaviors or characteristics of functions.
• They allow modifications to observe changes interactively.

Using these utilities is crucial for quickly determining properties like the domain, range, symmetry, and continuity of a function. In this case, it helps us easily apply the Horizontal Line Test by seeing the shape of \( h(x) \).

One-to-One Function

To understand what a one-to-one function is, imagine a function that passes the Horizontal Line Test. This means that for each output, there is a unique input. If a horizontal line cuts through the graph at more than one point, the function is not one-to-one.

• A function is one-to-one if it has exactly one distinct output for each input.
• This property is important as it allows the function to have an inverse.

For the sample function given, \( h(x) = \sqrt{16-x^2} \), using a graphing utility reveals that horizontal lines can cut the graph more than once. Therefore, it fails the Horizontal Line Test and is not a one-to-one function.

Inverse Function

Inverse functions are the 'reverse' of the original function. They swap the roles of inputs and outputs. If \( f(x) \) is our function, its inverse, denoted as \( f^{-1}(x) \), will satisfy the condition \( f(f^{-1}(x)) = x \).

• An inverse function reverses the effect of the original.
• A fundamental requirement for an inverse to exist is that the original must be one-to-one.

However, because \( h(x) = \sqrt{16-x^2} \) is not one-to-one, it means there cannot exist an inverse for it over its entire domain as originally defined. Understanding the inverse function concept relies heavily on grasping how the original function behaves across its range.