Problem 102 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the function \(f,\) (b) use the draw inverse feature of the graphing utility to draw the inverse relation of the function, and (c) determine whether the inverse relation is an inverse function. Explain your reasoning. $$f(x)=x \sqrt{4-x^{2}}$$

Step-by-Step Solution

Answer

The inverse relation of the function \(f(x) = x \sqrt{4-x^2}\) is not a function since it fails the vertical line test.

Step 1: Understanding the Function

The given function is \(f(x) = x \sqrt{4-x^2}\). You must first understand how this function behaves on a graph. It's a product of \(x\) and the square root of \((4-x^2)\), which is a parabola with a domain of \(-2 \leq x \leq 2\).

Step 2: Graphing the Function

Using a graphing utility, you can input the function \(f(x) = x \sqrt{4-x^2}\) to generate its graph. The graph should resemble a semicircle with endpoints at (-2,0) and (2,0).

Step 3: Drawing the Inverse Relation

The inverse relation can be graphed by reflecting the original graph about the line \(y=x\). Most graphing utilities have a feature that allows you to do this automatically. The result will be a graph that mirrors the original along the line \(y=x\).

Step 4: Determine If the Inverse is a Function

To determine whether the inverse relation is a function, apply the vertical line test. If a vertical line intersects the graph at more than one point, then the graph is not a function. In this case, the inverse relation is not a function because a vertical line would intersect the graph at more than one point.

Key Concepts

Graphing Utilities

Graphing utilities are valuable tools for visualizing mathematical functions and their behaviors. They allow you to input specific functions and view their graphs on various devices, such as graphing calculators or computer software. With these utilities, you can easily see the shape, intercepts, and other characteristics of the function without manually plotting multiple points.

In the exercise, you used a graphing utility to graph the function \( f(x) = x \sqrt{4-x^2} \). The graph helps to understand the visual representation, which looks like a semicircle with endpoints at (-2, 0) and (2, 0). It provides a clear insight into how the function behaves across its domain.

Another feature of graphing utilities is drawing inverse relations. This process involves reflecting the graph of the function across the line \( y = x \). This reflection helps in seeing what the inverse relation would look like. When working with inverse functions, understanding how to use these utility features can greatly simplify the process.

Vertical Line Test

The vertical line test is a simple method to determine whether a relation is a function. A function is defined as a relation where each input (or \( x \)-value) corresponds to exactly one output (or \( y \)-value).

When using the vertical line test, imagine drawing vertical lines through different points along the graph of the relation. If any vertical line touches the graph at more than one point, the relation is not a function. This is because it means an \( x \)-value has multiple corresponding \( y \)-values, violating the definition of a function.

In the given exercise, after reflecting the original function around the line \( y = x \) to plot the inverse, a vertical line crossed the graph at multiple points. Therefore, this tells us that the inverse relation is not a function, emphasizing the importance of the vertical line test in studying inverses.

Domain and Range

Understanding the domain and range is crucial for graphing and analyzing functions. The domain of a function represents all the possible input values (\( x \)-values) for which the function is defined. The range, on the other hand, consists of all the possible output values (\( y \)-values) the function can produce.

For the function \( f(x) = x \sqrt{4-x^2} \), the domain is \(-2 \leq x \leq 2\) because within this range, the expression under the square root is non-negative, allowing the function to be real-valued. This constraint arises from the need to take the square root of a non-negative number.

The range of this semicircle-shaped function graph is \(-2 \leq y \leq 2\). This corresponds to the vertical spread of the graph on the \( y \)-axis. Understanding these parameters helps in accurately using graphing utilities and ensuring the graphs are interpreted correctly. In the inverse relation, these values often switch places, which highlights how deeply intertwined domain and range are in function analysis.