Question

In Exercises \(17-20\), use a graphing utility to graph the function. Determine whether the function is one-to-one on its entire domain. $$ h(s)=\frac{1}{s-2}-3 $$

Short answer

The graph of the function \(h(s) = \frac{1}{s-2} - 3\) is such that no horizontal line crosses the function's graph at more than one point, hence it is one-to-one on its entire domain.

Step-by-step solution

Graph the function

Use a graphing utility to graph the function \(h(s)=\frac{1}{s-2}-3\). In general, the graph of this function has a vertical asymptote at \(s=2\). Also, it is shifted down by 3 units due to the -3 in the function.

Apply the Horizontal Line Test

The horizontal line test involves drawing a horizontal line that crosses the function's graph. If it touches the graph at more than one point, the function is not one-to-one. For \(h(s)\), this test needs to be applied across the entire domain of the function.

Determine if the function is one-to-one

After applying the horizontal line test to the graph, see if any horizontal line crosses the function's graph at more than one point. If it does, then the function is not one-to-one. If every horizontal line crosses the function's graph at most one point, then the function is one-to-one.

Key concepts

Graphing Functions

Graphing functions is an essential tool enabling students to visualize mathematical concepts and the behaviour of equations. One of the first steps in analyzing a function like \(h(s)=\frac{1}{s-2}-3\) is using a graphing utility, which aids in transforming the abstract formula into a concrete visual representation.

When graphing, we pay attention to various features such as intercepts, where the function crosses the axis, and the overall shape and direction of the graph. For the given function, graphing it reveals important characteristics such as a shift down by 3 units. The visual graph helps us understand the function's behaviour, and it is particularly valuable in determining its uniqueness, which leads us to use the horizontal line test.

Horizontal Line Test

The horizontal line test is a simple yet powerful method to determine if a function is one-to-one. When a horizontal line intersects a function's graph more than once, it implies that the function will produce the same output for different inputs, disproving its one-to-one nature.

For \(h(s)\), we conduct this test by visualizing if we could draw any horizontal line that crosses the graph multiple times. If such a situation arises, the function is not one-to-one. If, however, each horizontal line intersects the graph at most at one point, the function passes the test, confirming it as one-to-one. Therefore, applying the horizontal line test for the entire domain helps to ensure that for each output, there's only one corresponding input, which is a crucial condition for functions in certain mathematical and real-world applications.

Asymptotes

An asymptote is a line that a function's graph approaches but never touches or crosses. Asymptotes can be vertical, horizontal, or oblique, and they provide insight into the behaviour of a function as its inputs grow large or approach certain values.

In our exercise, the function \(h(s)=\frac{1}{s-2}-3\) has a vertical asymptote at \(s=2\). This tells us that as the input values, s, get closer to 2, the output values diverge towards infinity or negative infinity. The concept of asymptotes is important when graphing because it describes the function's behaviour near these critical values and can impact the range and domain of the function-understandings vital for further analysis in calculus and other fields of mathematics.