Problem 15 - Algebra 2 and Trigonometry

Question

In \(11-16,\) determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function. $$ \\{(-1,3),(-1,5),(-2,7),(-3,9),(-4,11)\\} $$

Step-by-Step Solution

Answer

There is no inverse because the given set is not a function; \(-1\) has two different outputs.

Step 1: Check for Unique Outputs

To determine if the function has an inverse, we first need to see if each input in the function pairs has only one unique output. Inspect the given set of pairs \{(-1,3),(-1,5),(-2,7),(-3,9),(-4,11)\}. Notice that the input \(-1\) is associated with two different outputs \(3\) and \(5\).

Step 2: Determine Function Type

Since \(-1\) is paired with both \(3\) and \(5\), this violates the definition of a function where each input must have a unique output. As this is not a proper function, it cannot have an inverse. For a function to have an inverse, it must be one-to-one, which means it passes the horizontal line test where each horizontal line intersects the graph at most once.

Step 3: Conclusion

Given the presence of repeated inputs with different outputs, the set is not a valid function. Consequently, it does not have an inverse because only bijective (one-to-one and onto) functions have inverses.

Key Concepts

Understanding Function Pairs

In the world of mathematics, function pairs are essentially input-output combinations or ordered pairs. They are like special mailboxes where one specific key (input) perfectly opens one unique compartment that contains just one letter (output). This consistent relationship is crucial as it determines how functions work.

• Each pair consists of a first element (the input or x-value) and a second element (the output or y-value).
• For a valid function, every key, or input \(x\), should only open one specific mailbox, showing a consistent and unique output \(y\).

If a function satisfies this condition, it behaves predictably, allowing you to know exactly what output follows when an input is selected. When each input produces exactly one output, we can comfortably talk about its inverse, which is like reversing the process to determine the original input from a given output.

Defining a One-to-One Function

A one-to-one function is a special type of function where each output value is associated with a distinct input value. This uniqueness makes it possible for the function to have an inverse.
To better comprehend one-to-one functions, consider the following attributes:

• Each input \(x\) corresponds to exactly one unique output \(y\).
• No two different inputs can produce the same output. In other words, if \(f(a) = f(b)\), then \(a = b\).
• One-to-one functions ensure that each mailbox has its own unique key that opens it.

The importance of one-to-one functions lies in their ability to be reversed. The ability to reliably switch \(x\) and \(y\) in any pair implies that each \(y\) also maps back to exactly one distinct \(x\). This reversibility is what makes inverse functions possible, making each input-output relationship a two-way street.

The Horizontal Line Test

The horizontal line test is a helpful visual method to determine if a function is one-to-one. It simplifies identifying whether a function, given as a graph, passes the rules necessary to have an inverse function.
Here's how the horizontal line test works:

• Draw horizontal lines across the graph of a function.
• If any horizontal line intersects the graph more than once, the function fails the test.
• For a function to be one-to-one, each horizontal line should intersect the function graph at most once.

By applying the horizontal line test, we can quickly and easily determine whether a function maintains the property of one-to-one correspondence in its domain. Upon passing this test, the conclusion can be drawn that an inverse is possible since each input and output pairing remains unique, ensuring a reversible relationship.