Problem 47 - 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. $$h(x)=\frac{x^{2}}{x^{2}+1}$$

Step-by-Step Solution

Answer

Yes, the function \(h(x) = \frac{x^2}{x^2 + 1}\) is one-to-one and so it has an inverse function.

Step 1: Graph the Function

Using a graphing utility, graph the function \(h(x) = \frac{x^2}{x^2 + 1}\). Observe the shape and where the curve of the graph lies in the coordinate plane.

Step 2: Apply the Horizontal Line Test

Draw horizontal lines through the graph. If any of these lines intersect the graph more than once, the function fails the Horizontal Line Test, meaning it is not one-to-one.

Step 3: Determine Whether the Function is One-to-One

For the function \(h(x) = \frac{x^2}{x^2 + 1}\), all horizontal drawn lines intersect the graph at one point only. Hence, the function is one-to-one.

Step 4: Determine Whether the Function has an Inverse

Since we determined that the function is one-to-one, it means that the function \(h(x) = \frac{x^2}{x^2 + 1}\) has an inverse. Any function that passes the Horizontal Line Test has an inverse function.

Key Concepts

Inverse Function

An inverse function is a fascinating concept in mathematics. If you have a function, say \( f(x) \), its inverse, denoted as \( f^{-1}(x) \), reverses the effect of \( f(x) \). When you apply \( f(x) \) to a value and then its inverse \( f^{-1}(x) \) to the output, you end up back at the original number you started with. Think of it like retracing your steps.
For instance, if \( f(a) = b \), then \( f^{-1}(b) = a \). The key here is this relationship must work for all values within the function's respective domains. That is why you need a function to be one-to-one to ensure each output is uniquely paired with one input.
This guarantees that a function can have an inverse since no two inputs can map to the same output.

One-to-One Function

A one-to-one function is essential when discussing inverses. A function is one-to-one if, for any two different inputs, the outputs are different. In simpler terms, every '\( x \)' in the domain corresponds to a unique 'output,' or '\( y \).' This property ensures that each output has only one input associated with it.

• Visual tool: The Horizontal Line Test helps identify one-to-one functions.
• If a horizontal line intersects the graph of the function more than once, the function is not one-to-one.
• For example, the function \( h(x)=\frac{x^2}{x^2+1} \) was found to be one-to-one using this test.

If a function is not one-to-one, it cannot have an inverse. This is because multiple inputs could map to a single output, making it impossible to determine the input from the output.

Graphing Utility

Graphing utilities are powerful tools in mathematics. These utilities allow us to visualize mathematical functions quickly and accurately. By graphing the function, we can see the complete behavior, such as the shape and position, which might not be easily inferred from the equation itself.
For example, with \( h(x)=\frac{x^2}{x^2+1} \), using a graphing utility helped observe that horizontal lines only intersect the graph at one point. This visualization confirmed that the function is one-to-one, which might not be immediately obvious from calculation alone.
Such tools are especially handy as they offer visual confirmation of properties like one-to-one behavior, making them indispensable in many math problems: