Problem 2
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. \(g=\\{(8,6),(9,6),(3,4),(-4,4)\\}\)
Step-by-Step Solution
Verified Answer
The function is not one-to-one, so it does not have an inverse.
1Step 1: Understand the Definition of One-to-One Function
A function is considered one-to-one if every element in the range is mapped from a distinct element of the domain. In other words, no two different inputs can map to the same output.
2Step 2: Check for Distinct Outputs in Mapping
Check the given set of pairs in the function \( g = \{(8,6),(9,6),(3,4),(-4,4)\} \). Observe that both 8 and 9 are mapping to 6, and 3 and -4 are mapping to 4. This shows that different inputs map to the same output.
3Step 3: Conclude if Function is One-to-One or Not
Based on the observation from the previous step, since there are pairs \((8,6)\) and \((9,6)\), and \((3,4)\) and \((-4,4)\) sharing the same output values, the function is not one-to-one.
4Step 4: Determine Inverse Applicability
Since the function is not one-to-one, it does not have an inverse that can be determined by switching inputs and outputs. Switching would not result in a valid function, as it would not pass the vertical line test.
Key Concepts
Inverse FunctionFunction MappingDomain and Range
Inverse Function
An inverse function is a fundamental concept in mathematics that essentially reverses the process of a function. If you have a function that maps input values (or domain) to output values (or range), the inverse function maps the outputs back to the inputs. However, not all functions have inverses. For a function to have an inverse, it must be a one-to-one function.
A one-to-one function ensures that each output is produced by exactly one input. If multiple inputs yield the same output, as seen in the example function \( g = \{(8,6),(9,6),(3,4),(-4,4)\} \), an inverse cannot exist. Attempting to reverse such a function will result in ambiguity because there would be more than one possible input for some outputs.
To verify if a function has an inverse, check if it's one-to-one. Once confirmed, you can find the inverse by swapping each pair's coordinates—switching inputs and outputs. However, since \( g \) is not one-to-one, it lacks a valid inverse.
A one-to-one function ensures that each output is produced by exactly one input. If multiple inputs yield the same output, as seen in the example function \( g = \{(8,6),(9,6),(3,4),(-4,4)\} \), an inverse cannot exist. Attempting to reverse such a function will result in ambiguity because there would be more than one possible input for some outputs.
To verify if a function has an inverse, check if it's one-to-one. Once confirmed, you can find the inverse by swapping each pair's coordinates—switching inputs and outputs. However, since \( g \) is not one-to-one, it lacks a valid inverse.
Function Mapping
Function mapping refers to the process where each element from the domain is paired with an element in the range. It's analogous to assigning a unique name from a roster to a unique seat in an auditorium.
In the context of the function \( g = \{(8,6),(9,6),(3,4),(-4,4)\} \), function mapping shows how elements of the domain (inputs) \( 8, 9, 3, \) and \( -4 \) are paired with elements of the range (outputs) \( 6 \) and \( 4 \).
A clear and precise function mapping helps in understanding the behavior of functions, especially in determining if a function is one-to-one. When different inputs are mapped to the same output, as in the function \( g \), it indicates that the function is not one-to-one, and thus cannot be an inverse function.
In the context of the function \( g = \{(8,6),(9,6),(3,4),(-4,4)\} \), function mapping shows how elements of the domain (inputs) \( 8, 9, 3, \) and \( -4 \) are paired with elements of the range (outputs) \( 6 \) and \( 4 \).
A clear and precise function mapping helps in understanding the behavior of functions, especially in determining if a function is one-to-one. When different inputs are mapped to the same output, as in the function \( g \), it indicates that the function is not one-to-one, and thus cannot be an inverse function.
Domain and Range
Understanding domain and range is crucial in characterizing and working with functions. The domain of a function represents all the possible input values, while the range represents all potential output values.
In the given function \( g = \{(8,6),(9,6),(3,4),(-4,4)\} \), the domain is the set \{8, 9, 3, -4\} and the range is \{6, 4\}.
Inspecting the domain and range gives insight into the nature of the function. It helps in determining if the function is one-to-one. If two different domain values, like \( 8 \) and \( 9 \), produce the same range value, such as \( 6 \), it indicates the function is not one-to-one. A function with a clear, unique mapping from domain to range—where each domain value maps to a unique range value—ensures it has a potential inverse.
In the given function \( g = \{(8,6),(9,6),(3,4),(-4,4)\} \), the domain is the set \{8, 9, 3, -4\} and the range is \{6, 4\}.
Inspecting the domain and range gives insight into the nature of the function. It helps in determining if the function is one-to-one. If two different domain values, like \( 8 \) and \( 9 \), produce the same range value, such as \( 6 \), it indicates the function is not one-to-one. A function with a clear, unique mapping from domain to range—where each domain value maps to a unique range value—ensures it has a potential inverse.
Other exercises in this chapter
Problem 2
Use a calculator to approximate each logarithm to four decimal places. $$ \log 6 $$
View solution Problem 2
For the functions \(f\) and \(g\), find a. \((f+g)(x)\), b. \((f-g)(x), c .(f \cdot g)(x)\), and d. \(\left(\frac{f}{g}\right)(x)\). $$ f(x)=x+4 ; g(x)=5 x-2 $$
View solution Problem 2
Graph each exponential function. $$ y=4^{x} $$
View solution Problem 2
Write each sum as a single logarithm. Assume that variables represent positive numbers. $$ \log _{3} 8+\log _{3} 4 $$
View solution