Problem 17 - Calculus of a Single Variable

Question

Use a graphing utility to graph the function. Then use the Horizontal Line Test to determine whether the function is one-to-one on its entire domain and therefore has an inverse function. \(h(s)=\frac{1}{s-2}-3\)

Step-by-Step Solution

Answer

To determine if the function \(h(s) = \frac{1}{s-2}-3\) is one-to-one and hence, has an inverse, you need to graph it using a graphing utility and then apply the Horizontal Line Test. If every horizontal line intercepts the graph at most one point, the function is one-to-one and has an inverse function.

Step 1: Graph the Function

Firstly, graph the function \(h(s) = \frac{1}{s-2}-3\) using a graphing utility. This graph will serve as the visual representation needed for the subsequent step.

Step 2: Apply Horizontal Line Test

After graphing the function, apply the Horizontal Line Test. This test determines if a function is one-to-one and involves drawing horizontal lines through the graph. If each horizontal line touches the graph at most one point, the function passes the test, meaning it's one-to-one.

Step 3: Determine if Function has an Inverse

From the results of the Horizontal Line Test, determine whether the function is one-to-one on its entire domain (i.e., if it passes the test). If it does, the function has an inverse.

Key Concepts

Horizontal Line Test

The Horizontal Line Test is a crucial method for determining whether a function is one-to-one. This test is simple: imagine drawing horizontal lines across a graph. If any horizontal line intersects the graph more than once, the function fails the test. This means it's not a one-to-one function. Passing the Horizontal Line Test guarantees that each output value in the function is unique to one input value.

This is important because for a function to have an inverse, it must be one-to-one. In the case of the function \(h(s) = \frac{1}{s-2} -3\), if we find that no horizontal line intersects the graph more than once, the function has an inverse. Applying this test visually using a graph helps to quickly identify the uniqueness of the function's output values.

One-to-One Functions

A function is defined as one-to-one if every input is paired with a unique output, with no duplicates. This means for every \(x\) value, there is a unique \(y\) value and vice versa. One-to-one functions are special because they always have an inverse function. This inverse "undoes" the effect of the original function, reversing its operations.

In mathematical terms, a function \(f(x)\) is one-to-one if \(f(a) = f(b)\) implies that \(a = b\). In other words, if two different inputs produce the same output, the function fails to meet the one-to-one criteria. For functions like \(h(s) = \frac{1}{s-2} -3\), being one-to-one ensures that there exists a function that can reverse its operations, thus guaranteeing an inverse.

Graphing Utilities

Graphing utilities are powerful tools used to visualize functions and perform analysis such as the Horizontal Line Test. These utilities, which might include software like graphing calculators or online graphing apps, help create accurate visual graphs of functions.

With functions like \(h(s) = \frac{1}{s-2} -3\), graphing utilities provide a clear picture of how the function behaves across its domain. Being able to visualize the graph aids in applying tests like the Horizontal Line Test. Additionally, graphing utilities help to find key features of the graph, such as intercepts, asymptotes, and intervals of increase or decrease.

• These tools save time by quickly and accurately plotting complex functions.
• They support deeper comprehension of functions through visual representation.
• Interactive features often allow for manipulation and exploration of the graph.

Incorporating graphing utilities into your math study routine can significantly enhance your understanding and analytical skills.