Problem 6
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. \(g=\\{(0,3),(3,7),(6,7),(-2,-2)\\}\)
Step-by-Step Solution
Verified Answer
The function is not one-to-one, so it does not have an inverse.
1Step 1: Check for One-to-One
To determine if the function is one-to-one, each output value must come from a unique input value. Examine the given function \(g=\{(0,3),(3,7),(6,7),(-2,-2)\}\). Notice that the output '7' is associated with two different inputs (3 and 6). Therefore, this function is not one-to-one since it does not satisfy the criteria for unique outputs.
2Step 2: Conclude Based on One-to-One Check
Since the function is not one-to-one (as shown by the repeated output value '7' for two different inputs), it does not have an inverse function that can be determined by switching inputs and outputs.
Key Concepts
Inverse FunctionsUnique OutputsInput-Output RelationshipsFunction Properties
Inverse Functions
In mathematics, an inverse function is essentially a way of reversing the process of an original function. Think of it like having a magical mirror that shows you how to undo the effects of a function. When you have a function that is one-to-one, each input maps to a unique output, and if you reverse this relationship, you end up with the inverse function. For example, if a function maps '2' to '5', its inverse will map '5' back to '2'. This idea is captured by the notation \(f^{-1}(x)\), where \(f\) is the original function and \(f^{-1}\) is its inverse.
However, it's crucial to understand that not all functions have inverses. A function must be one-to-one in order for an inverse to exist, because if two different inputs map to the same output, there's no way of determining which input to "undo" to. Thus, before finding an inverse, checking if the function is one-to-one is the first key step.
However, it's crucial to understand that not all functions have inverses. A function must be one-to-one in order for an inverse to exist, because if two different inputs map to the same output, there's no way of determining which input to "undo" to. Thus, before finding an inverse, checking if the function is one-to-one is the first key step.
Unique Outputs
To determine if a function is one-to-one, an essential property to check is the uniqueness of its outputs. If every output in the set of ordered pairs is associated with a distinct input, then the function can be considered one-to-one. In simple terms, think of it as checking if every output connects to exactly one input.
For example, consider the function shown as\(g=\{(0,3),(3,7),(6,7),(-2,-2)\}\). Here, observe the output '7' appears more than once, linking to both '3' and '6'. This violates the condition for unique outputs. It's like having two different keys (inputs) that open the same lock (output). As such, this lack of uniqueness in outputs implies that the function \(g\) is not one-to-one and consequently, does not possess an inverse function.
For example, consider the function shown as\(g=\{(0,3),(3,7),(6,7),(-2,-2)\}\). Here, observe the output '7' appears more than once, linking to both '3' and '6'. This violates the condition for unique outputs. It's like having two different keys (inputs) that open the same lock (output). As such, this lack of uniqueness in outputs implies that the function \(g\) is not one-to-one and consequently, does not possess an inverse function.
Input-Output Relationships
The relationship between inputs and outputs is at the heart of understanding whether a function is one-to-one. In a function, each input should lead to a specific output, demonstrating a clear path or direction. In the context of a one-to-one function, every distinct input corresponds to a distinct output.
This relationship is critical because if the path from input to output isn't unique, it becomes difficult to reverse it. Suppose we have an input-output grouping for function \(g\), like \(g=\{(0,3),(3,7),(6,7),(-2,-2)\}\). If more than one path leads to the same output, reversing the relationship (finding an inverse) isn't feasible. The output '7' here is a perfect example, as it is a product of two different inputs, disrupting the possibility of a unique detour back—hence no inverse.
This relationship is critical because if the path from input to output isn't unique, it becomes difficult to reverse it. Suppose we have an input-output grouping for function \(g\), like \(g=\{(0,3),(3,7),(6,7),(-2,-2)\}\). If more than one path leads to the same output, reversing the relationship (finding an inverse) isn't feasible. The output '7' here is a perfect example, as it is a product of two different inputs, disrupting the possibility of a unique detour back—hence no inverse.
Function Properties
Understanding the properties of functions is key when analyzing them for one-to-one nature and potential inverses. Key properties to consider include the domain, range, and the uniqueness of the input-output pairings.
- The **Domain** is the set of all possible inputs a function can take. In relation to function \(g\), it includes \(0, 3, 6,\) and \(-2\).
- The **Range** is the set of all possible outputs, indicating what results come from the function operation. For \(g\), the range consists of the values \(3, 7,\) and \(-2\).
- The **One-to-One Property** requires each input to have a different output and vice versa. A vital check here is ensuring no duplicated outputs for different inputs.
Other exercises in this chapter
Problem 6
Use a calculator to approximate each logarithm to four decimal places. See Examples I and 5. \(\ln 3\)
View solution Problem 6
Write each sum as a single logarithm. Assume that variables represent positive numbers. See Example 1 . $$ \log _{5} y^{3}+\log _{5}(y-7) $$
View solution Problem 6
For the functions \(f\) and \(g,\) find \(a .(f+g)(x), b .(f-g)(x)\) \(c .(f \cdot g)(x),\) and \(d .\left(\frac{f}{g}\right)(x) .\) $$f(x)=\sqrt[3]{x}, g(x)=x-
View solution Problem 7
Write each as an exponential equation. See Example 1. $$ \log _{9} x=4 $$
View solution