Problem 38 - Algebra and Trigonometry Real Mathematics, Real People

Question

Does the function have an inverse? Explain. $$\\{(2,4),(3,7),(7,2)\\}$$

Step-by-Step Solution

Answer

Yes, the function does have an inverse.

Step 1: Identify the ordered pairs in the function.

The given function is represented as ordered pairs. The pairs are (2, 4), (3, 7), and (7, 2).

Step 2: Check if all the first elements (x-values) are distinct.

Checking the x-values (2, 3, 7), they are distinct. This means that each input gives a unique output.

Step 3: Check if all the second elements (y-values) are distinct.

The y-values are 4, 7, and 2. Like the x-values, these are also distinct, indicating that each output corresponds to one unique input.

Step 4: Conclusion.

Because all x-values (inputs) and y-values (outputs) are distinct, we can say that the function does have an inverse.

Key Concepts

Function

A function is a relationship between a set of inputs and a set of outputs. Each input (often thought of as an "x-value") is associated with one specific output (a "y-value"). In a function, no input can correspond to more than one output. This is a fundamental property that different mathematical functions adhere to. Functions can be represented in various ways, such as equations, graphs, or sets of ordered pairs. Understanding the nature of functions is vital for determining whether certain operations, like finding an inverse, are possible. Functions that map inputs to precisely one output and not multiple do pave the way for finding inverses, which essentially reverses the direction of the function from input-to-output to output-to-input.

Ordered Pairs

Ordered pairs are a method used to express functions and relations. An ordered pair consists of two elements: the first element is typically an "x-value," and the second is a "y-value." For example, in the ordered pair \((2, 4)\), 2 is linked to 4. When ordered pairs are placed in curly braces, they represent a function. In this context, a set of ordered pairs shows which inputs map to which outputs.

• For the pair (2, 4), 2 is mapped to 4.
• For the pair (3, 7), 3 is mapped to 7.
• For the pair (7, 2), 7 is mapped to 2.

This set of ordered pairs provides a clear view of the function's structure, helping to verify if each input links uniquely to a singular output.

Distinct x-values

When evaluating if a function possesses an inverse, ensuring that all first elements—the x-values—in the ordered pairs are distinct is crucial. If the x-values of a function are distinct, that means each x-value leads to a unique y-value. Considering our example's set of ordered pairs, \((2, 4), (3, 7), (7, 2)\):

• The x-values are 2, 3, and 7.

Each of these inputs is different, leaving no overlap, which meets the necessary condition required for the possibility of the function having an inverse. Without distinct x-values, it would be impossible for each output to have only one path back to its input in the inverse process.

Distinct y-values

Distinct y-values mean that each output value is unique and corresponds to a single input value. For a function to have an inverse that is also a function, the y-values must be distinct. This ensures that each output leads back to one unique input.In our specific set of ordered pairs, \((2, 4), (3, 7), (7, 2)\), the y-values, or outputs, are also 4, 7, and 2, all of which are distinct.

• Output 4 corresponds uniquely to input 2.
• Output 7 corresponds uniquely to input 3.
• Output 2 corresponds uniquely to input 7.

With these unique correspondences, we affirm that the function meets the criteria necessary for it to have an inverse. Each y-value leading uniquely back to one x-value is key to reversing the function back to its original form.