Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Does the function have an inverse? Explain. $$\\{(-3,6),(-1,5),(0,6)\\}$$

Step-by-Step Solution

Answer

No, the function does not have an inverse because it's not one-to-one.

Step 1: Identify the function's output values

First, identify all the output values, or y-values, from the given set of ordered pairs: \(6, 5, 6\).

Step 2: Check for repeated output values

Determine if there are any repeated output (y) values. In this case, the number \(6\) appears twice in the output values.

Step 3: Determine if the function has an inverse

The function doesn't have an inverse if there are any repeating output values. Since number \(6\) is repeated, this function does not have an inverse.

Key Concepts

Ordered Pairs

Ordered pairs are the foundation of many mathematical concepts, especially when discussing functions. An ordered pair is a set of two elements, typically represented as \((x, y)\). These elements are usually numbers, where each pair is a specific combination of inputs and outputs. In the function example \(\{(-3,6),(-1,5),(0,6)\}\), each ordered pair consists of a first number (input or x-value) and a second number (output or y-value).
When working with ordered pairs, the order of the numbers is crucial. The first position typically represents the input and the second represents the output. It’s important to maintain this order as it affects the relationship being described.
In terms of functions, a set of ordered pairs is used to define the relationship between input and output values. Every function can be visualized as a collection of such pairs.

Function Outputs

In the context of functions, the output, or y-values, represents the results you get after applying the function rule to the inputs. In \(\{(-3,6),(-1,5),(0,6)\}\), the outputs are \(6, 5, 6\). These are considered the y-values of each ordered pair. Understanding function outputs is essential because it determines whether a function can have an inverse. The outputs are the values that the inputs map onto. If no output is repeated, then each input maps to a unique output, which is a crucial condition for the existence of an inverse function.
Generally, to find all the output values of a function:

• Identify the y-value from each ordered pair.
• List each output value only once to see if any value repeats.

Repeated outputs impact whether a function has an inverse, which brings us to our next important concept.

Repeated Values

Repeated values in the outputs, or y-values, are significant when analyzing functions to determine if an inverse exists. In a set of ordered pairs like \(\{(-3,6),(-1,5),(0,6)\}\), the output value \(6\) repeats itself, appearing in both the first and third pairs.A function is invertible, meaning it has an inverse, only if every output value in the set is unique. This uniqueness ensures that each output corresponds to only one input. If there's any repetition, like in the case of \(6\) appearing twice, the function cannot have an inverse.
This rule exists because the inverse function, which swaps the roles of inputs and outputs, would not be a true function if it included multiple inputs mapping to a single output. So, when checking for inverses:

• Look for any repeated y-values among the ordered pairs.
• If any y-value repeats, the function does not have an inverse.

A function with repeated values like the given example proves the point and shows why attention to y-values is necessary to determine invertibility.