Problem 3 - Intermediate Algebra

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

Step 1: Define One-to-One Function

A function is considered one-to-one if each element of the range corresponds to exactly one element of the domain.

Step 2: Analyze the Given Function

The given function is a set with one ordered pair: \( h = \{(10,10)\} \). This means the input 10 is mapped to the output 10.

Step 3: Check One-to-One Condition

Since there's only one pair and no repeated input or output, every element of the output (range) corresponds to exactly one element of the input (domain). Thus, it satisfies the one-to-one condition.

Step 4: Find the Inverse

To find the inverse of a one-to-one function, swap the coordinates of each pair. For \( h = \{(10,10)\} \), swapping input and output gives \( h^{-1} = \{(10,10)\} \). This is because both input and output are identical.

Key Concepts

Inverse Function

An inverse function essentially reverses the original function's operations. If we have a function \( f(x) \), then its inverse, \( f^{-1}(x) \), will undo the action of \( f(x) \). To find an inverse, it's crucial that the function is one-to-one, meaning each output is uniquely linked to a single input. This exclusivity ensures that the role of inputs and outputs can be reversed without ambiguity.

For the function given as \( h = \{(10,10)\} \), it defines a simple one-to-one mapping where the input and output value are identical. Swapping the coordinates thus yields the same ordered pair, \( h^{-1} = \{(10,10)\} \). This unique aspect of identical input-output pairs makes the inverse quite intuitive as the function is inherently its own inverse.

Domain and Range

Understanding the domain and range of a function is like knowing the full capability of what the function can handle and produce. The domain is the set of all conceivable inputs, while the range is all possible outputs the function can yield. For a function to be one-to-one, the range must associate with only one particular input in the domain.

In the function \( h = \{(10,10)\} \), the domain contains just the number 10, and so does the range. This means there’s only one element in both sets, which simplifies analysis dramatically. Each input of 10 leads to output 10, fulfilling the one-to-one condition since there's just one input and one precise output.

Function Analysis

Analyzing functions encompasses checking their properties, like one-to-one characteristics, to understand they're properly defined and find details like their inverse. To confirm a function is one-to-one, observe every input-output pair: does each range element connect to precisely one domain element?

For our function \( h = \{(10,10)\} \), the analysis reveals its simplicity. With only one pair, \({10}\) mapped to \({10}\), there's clearly no repetition, and the mapping's uniqueness is uncontested. Determining the inverse involves the orderly swap of pairs, which here remains unchanged due to identical inputs and outputs—highlighting the function's self-inverse property with in-depth clarity.