Problem 3 - Algebra A Combined Function

Question

Determine whether each function is a one-to-one function. If it is one-to-one, list the inverse function by switching coordinates, or inputs and outputs. \(h=\\{(10,10)\\}\)

Step-by-Step Solution

Answer

The function \( h \) is one-to-one, and its inverse is \( \{(10, 10)\}.\)

Step 1: Define a One-to-One Function

A function is defined as one-to-one if each element in the domain maps to a unique element in the range. This implies that no two distinct inputs from the domain map to the same output in the range.

Step 2: Analyze the Given Function

The given function is described by the set \( h = \{(10, 10)\} \). This means that the input 10 maps to the output 10, and there is only a single pair of corresponding input and output.

Step 3: Verify One-to-One Property

Since there is only one pair, there are no additional inputs that need to be checked for uniqueness. Therefore, the function \( h \) satisfies the definition of a one-to-one function because there isn't any possibility of a second different input mapping to the same output.

Step 4: Determine the Inverse

To find the inverse of a one-to-one function, we switch each input-output pair. For the pair \((10, 10)\), switching the input and the output gives us the pair \((10, 10)\). Therefore, the inverse of \( h \) is also \( \{(10, 10)\} \).

Key Concepts

Inverse Function

An inverse function essentially reverses the mapping of the original function. If you have a function that takes an input and produces an output, the inverse does the opposite—it takes that output and returns the original input. For a function to have an inverse, it must be one-to-one. When each input maps to a unique output and vice versa, switching them around gives you the inverse.For instance, if we have a function pair like \((a, b)\), its inverse switches the places of the input and output, resulting in \((b, a)\). In the example, the function \(h = \{(10, 10)\}\) already has input and output as the same, so its inverse is, perhaps predictably, just itself \(\{(10, 10)\}\). This switching maintains the integrity of the mappings, which is why this function qualifies as its own inverse.

Domain and Range

The domain and range of a function are fundamental concepts that define the model of input-output mapping. The domain represents all possible inputs for the function, while the range covers all possible outputs.For a one-to-one function, each element of the domain maps to a single unique element in the range and vice versa. If considering the function \(h = \{(10, 10)\}\), the domain is simply \(\{10\}\), and the range is also \(\{10\}\). Since the function is defined by one pair, the domain and range respectively comprise that single value, both ensuring the function fits the criteria for one-to-one characteristic.

Function Mapping

Mapping in functions refers to how inputs (from the domain) are associated with outputs (in the range). This concept is fundamental to understanding how functions operate. Think of it like a machine: you put something in, the function processes it, and something comes out.In a one-to-one function, this mapping needs to be neat and organized. Imagine if a function was a busy city road—each car (input) should travel down its own distinct lane and arrive at its unique destination (output).In our specific example \((10, 10)\), the mapping is straightforward, as one input maps to one output with no overlap or ambiguity. This clean mapping ensures that it's easy to find an inverse because each output leads back to exactly one input, reflecting the consistent and orderly nature of function mapping.