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Chapter 6: Problem 19

Determine whether the function is one-to-one. $$ \\{(0,0),(1,1),(2,16),(3,81)\\} $$

Short Answer

Expert verified
The function is one-to-one because all output values are unique.

Step by step solution

01

- Understanding the Definition of One-to-One Function

A function is one-to-one if each output value is unique; this means no two different input values map to the same output value.
02

- Listing the Given Pairs

The given function has the pairs: \((0,0), (1,1), (2,16), (3,81)\).
03

- Checking for Repeated Output Values

Look at the output values \(0, 1, 16, 81\) and check if any value is repeated. In this case, all values are unique.
04

- Conclusion

Since all output values are unique, the function is one-to-one.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Function Analysis
Function analysis is the process of examining the characteristics of a function. A function, in mathematics, is a relation between a set of inputs and a set of possible outputs where each input is related to exactly one output.
This means analyzing how each input value is paired with its output value.
For example, in the function \(\{(0,0),(1,1),(2,16),(3,81)\}\), each input (0, 1, 2, 3) is matched with a unique output (0, 1, 16, 81).
When we do function analysis, we check for properties like whether the function is one-to-one, continuous, or periodic.
What Are Unique Output Values?
In a function, output values are the results we get from plugging input values into the function.
Unique output values mean that no two different inputs share the same output.
For instance, in the function pairs \(\{(0,0),(1,1),(2,16),(3,81)\}\), the outputs are 0, 1, 16, and 81.
Since none of these values are repeated, they are unique.
Unique output values are crucial in determining if a function is one-to-one.
What is the Definition of a One-to-One Function?
A function is called one-to-one (or injective) if every output value corresponds to exactly one input value.
In other words, no output value is paired with more than one input value.
This is important because it ensures that the function is consistent and predictable.
For example, the function \(\{(0,0),(1,1),(2,16),(3,81)\}\) is one-to-one because each input is paired with a unique output.
To check if a function is one-to-one, we list the pairs and verify that no output value is repeated.
Understanding Function Pairs
Function pairs are sets of input and output values that define a function.
They are written as \( (input, output) \) pairs.
For example, the function \(\{(0,0),(1,1),(2,16),(3,81)\}\) consists of four pairs: (0,0), (1,1), (2,16), and (3,81).
These pairs show how each input maps to a specific output.
By examining function pairs, we can determine properties like whether the function is one-to-one, continuous, or periodic.
Analyzing these pairs is essential for understanding how a function behaves and for solving exercises correctly.

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