Problem 14 - 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. $$ \\{(2,7),(3,7),(4,7),(5,7),(6,7)\\} $$

Step-by-Step Solution

Answer

The function does not have an inverse because it is not one-to-one.

Step 1: Understand the Given Function

The given function is a set of ordered pairs: \((2,7), (3,7), (4,7), (5,7), (6,7)\). Here, each input value is associated with the same output value of 7.

Step 2: Evaluate One-to-One Property

A function must be one-to-one to have an inverse, meaning different inputs should produce different outputs. In this case, all inputs \(2, 3, 4, 5, 6\) produce the same output 7, indicating the function is not one-to-one.

Step 3: Confirm No Inverse Function

Since the function is not one-to-one, it does not satisfy the required condition for having an inverse function. Without a unique mapping from each input to a unique output, the inverse does not exist.

Key Concepts

One-to-One Function

A crucial property for a function to have an inverse is that it must be a one-to-one function. This means that each input value maps to a distinct and unique output value. With a one-to-one function:

• No two different inputs have the same output.
• Each output corresponds to exactly one input.

Consider a situation where two friends have identical names. If you call out, they both respond, creating confusion. Likewise, in functions, if multiple inputs lead to the same output, it's not one-to-one. For the given exercise, since all inputs (2, 3, 4, 5, and 6) lead to the output 7, the function is not one-to-one. Without this distinct mapping, an inverse function, which needs to reverse the process uniquely, cannot exist.

Ordered Pairs

Ordered pairs are fundamental building blocks of functions. They represent the input-output relationships that define the function's behavior. An ordered pair includes:

• The first element (typically called 'x') representing the input.
• The second element (typically called 'y') representing the output.

For instance, in the pair (2, 7), 2 is the input, and 7 is the output.In our exercise, the function is comprised of the set \[\{(2,7), (3,7), (4,7), (5,7), (6,7)\}\] This indicates that each input (2, 3, 4, 5, and 6) maps to the single output 7. If examined individually, each ordered pair suggests: "when the input is 2, the output is 7" and so forth.The lack of unique outputs from these ordered pairs prevents the function from being one-to-one, which is essential for the existence of an inverse.

Function Evaluation

Evaluating a function involves determining the output for given inputs based on predetermined rules or relationships. Functions can be represented by:

• Equations
• Graphs
• Sets of ordered pairs

In our exercise, evaluating the function involves examining the set \[\{(2,7), (3,7), (4,7), (5,7), (6,7)\}\] For any given input like 2, our function evaluation leads to an output of 7. Similarly, inputs 3, 4, 5, and 6 all produce the output 7. This consistent output is evidence that the function does not vary with different inputs, further reinforcing that it's not one-to-one.Therefore, while the function evaluation provides clarity on what outputs arise from specific inputs, the failure to produce unique outputs means no inverse function can be derived.