Problem 43 - Precalculus with Limits

Question

In Exercises 43-48, 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. \(g(x) = \frac{4-x}{6}\)

Step-by-Step Solution

Answer

The function \(g(x) = \frac{4-x}{6}\) is a one-to-one function, thus it has an inverse.

Step 1: Graph the function

Use a graphing software or calculator to plot the function \(g(x) = \frac{4-x}{6}\). The graph will be a straight line, as this equation is linear.

Step 2: Implement the Horizontal Line Test

Apply the Horizontal Line Test (HLT). This test helps to determine if a function possesses an inverse. If a horizontal line intersects the graph at only one point, the function proves to be a one-to-one function and it possesses an inverse.

Step 3: Interpret the results

Observe the graph. When applying the HLT, it appears that every horizontal line intersects the graph at exactly one point. This means the function is one-to-one and hence, possesses an inverse.

Key Concepts

Graphing Utility

Graphing utilities such as calculators and computer software are essential tools in visualizing mathematical functions, particularly when we need to understand their characteristics. To graph the given function, \( g(x) = \frac{4-x}{6} \), you'd typically enter the equation into a graphing utility and set an appropriate scale for both axes.

Here's a simplified approach on how to use a graphing utility effectively:

• Select a suitable view window that encompasses the graph.
• Enter the equation of the function in the utility's graphing interface.
• Analyze the graph that's plotted, paying close attention to slope and y-intercept, as these will help when conducting the Horizontal Line Test.

Understanding how to interpret the graph plot is crucial, as it lays the groundwork for determining whether a function is one-to-one.

One-to-One Function

A one-to-one function is a type of function where each output value is uniquely paired with only one input value. In other words, no two different inputs in a one-to-one function map to the same output.

Visual cues when graphing can be immensely helpful. With the Horizontal Line Test (HLT), you can easily identify a one-to-one function. If every horizontal line intersects with the graph at most once, then the function is declared one-to-one. For our function \( g(x) = \frac{4-x}{6} \), the graph is a straight line, meaning it will meet any horizontal line at exactly one point, satisfying the criteria for a one-to-one function.

Confirming that a function is one-to-one is not just a mathematical exercise; it has practical implications such as ensuring that each input has a unique and predictable output, which is essential in scenarios like cryptography, data storage, and even in day-to-day organizational tasks.

Inverse Function

The existence of an inverse function is tied directly to whether a function is one-to-one. An inverse function, essentially, undoes the action of the original function. For the linear function \( g(x) = \frac{4-x}{6} \), if it passes the Horizontal Line Test, which our function does, it assuredly has an inverse function.

To find the inverse, one would typically swap the \(x\) and \(y\) in the equation and then solve for \(y\). This process results in the inverse function, often denoted as \(g^{-1}(x)\), which will reflect the original function across the line \(y=x\). The graphical representation of an inverse function can be a mirror image of the original function's graph when flipped over the line \(y=x\), reaffirming their relationship.

Understanding inverse functions is critical because it allows us to 'reverse' computations and can provide insights into the original function's behavior. It's particularly helpful in solving equations and modeling real-world scenarios where reversing a process is required, such as in deconvolution in signal processing or finding original amounts before tax or interest.