Problem 8
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. $$ \begin{array}{|l|c|c|c|c|c|} \hline \text { State (Input) } & \text { Texas } & \text { Massachusetts } & \text { Nevada } & \text { Idaho } & \text { Wisconsin } \\ \hline \text { Number of Two-Year Colleges (0utput) } & 70 & 22 & 3 & 3 & 31 \\\ \hline \end{array} $$
Step-by-Step Solution
Verified Answer
The function is not one-to-one because Nevada and Idaho both yield the output of 3.
1Step 1: Identify the function's pairs
The function relates U.S. states to the number of two-year colleges. It can be expressed as a set of input-output pairs: \((\text{Texas}, 70)\), \((\text{Massachusetts}, 22)\), \((\text{Nevada}, 3)\), \((\text{Idaho}, 3)\), and \((\text{Wisconsin}, 31)\).
2Step 2: Analyze the outputs for uniqueness
Check if each output is associated with exactly one input. A function is one-to-one if each possible output is linked with a distinct input.
3Step 3: Determine function type
In the given pairs, the output 3 corresponds to two different inputs: Nevada and Idaho. Hence, the function is not one-to-one since it does not meet the requirement of having a unique output for each input.
Key Concepts
Inverse FunctionInput-Output PairsFunction Analysis
Inverse Function
An inverse function essentially swaps the roles of inputs and outputs in a given function. In mathematical terms, if we have a function \( f \) that maps \( x \) to \( y \), the inverse function, denoted as \( f^{-1} \), will map \( y \) back to \( x \). This can only happen if the function is one-to-one. Why? Because each output needs to correspond to just one singular input, ensuring that there's a clear path to reverse the process.
However, in instances like our exercise, where more than one input shares the same output (3 colleges for both Nevada and Idaho), we cannot define an inverse function. This is because you wouldn't know how to associate the output back to a unique input, thereby violating the essential property needed for an inverse function. So remember, identifying whether a function is one-to-one is critical before determining its inverse.
However, in instances like our exercise, where more than one input shares the same output (3 colleges for both Nevada and Idaho), we cannot define an inverse function. This is because you wouldn't know how to associate the output back to a unique input, thereby violating the essential property needed for an inverse function. So remember, identifying whether a function is one-to-one is critical before determining its inverse.
Input-Output Pairs
To understand functions, noting their input-output pairs is fundamental. These pairs tell us how a specific input is transformed to an output. For our exercise, each state is an input, while the number of two-year colleges represents the output. We have pairs such as:
By organizing and analyzing these input-output pairs, you get insights into the uniqueness of the mapping. It becomes simple yet powerful in function analysis, helping determine other properties, including the potential existence of an inverse function.
- Texas mapped to 70
- Massachusetts mapped to 22
- Nevada and Idaho both mapped to 3
- Wisconsin mapped to 31
By organizing and analyzing these input-output pairs, you get insights into the uniqueness of the mapping. It becomes simple yet powerful in function analysis, helping determine other properties, including the potential existence of an inverse function.
Function Analysis
Function analysis involves breaking down a function's characteristics to understand its behavior. In this context, it's about distinguishing whether a given function is one-to-one.
Firstly, check each pair to see if every output corresponds to a single input. In the given exercise, Nevada and Idaho share the same output (3), making the function fail the one-to-one test. This analysis process is essential because a function that is not one-to-one cannot have an inverse.
Moreover, this process enriches your understanding of how functions work and prepares you to tackle more complex problems effectively. Through function analysis, uncover patterns and properties that guide subsequent steps, ensuring clarity on how inputs are transformed into outputs. Being methodical in this approach ensures we avoid mistakes in determining the characteristics of a function, such as establishing if it has an inverse. Function analysis is an indispensable tool for anyone involved in mathematical problem-solving.
Firstly, check each pair to see if every output corresponds to a single input. In the given exercise, Nevada and Idaho share the same output (3), making the function fail the one-to-one test. This analysis process is essential because a function that is not one-to-one cannot have an inverse.
Moreover, this process enriches your understanding of how functions work and prepares you to tackle more complex problems effectively. Through function analysis, uncover patterns and properties that guide subsequent steps, ensuring clarity on how inputs are transformed into outputs. Being methodical in this approach ensures we avoid mistakes in determining the characteristics of a function, such as establishing if it has an inverse. Function analysis is an indispensable tool for anyone involved in mathematical problem-solving.
Other exercises in this chapter
Problem 8
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Graph each exponential function. $$ y=\left(\frac{1}{3}\right)^{x}+2 $$
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Write each sum as a single logarithm. Assume that variables represent positive numbers. $$ \log _{6} 3+\log _{6}(x+4)+\log _{6} 5 $$
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