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Chapter 6: Problem 20

Determine whether the function is one-to-one. $$ \\{(1,2),(2,8),(3,18),(4,32)\\} $$

Short Answer

Expert verified
The function is one-to-one.

Step by step solution

01

- Understand a One-to-One Function

A function is one-to-one if and only if different inputs produce different outputs. In other words, for every pair of inputs \(x_1\) and \(x_2\), if \(x_1 鈮� x_2\), then \(f(x_1) 鈮� f(x_2)\).
02

- List All Input-Output Pairs

Write down the given pairs: \( (1,2), (2,8), (3,18), (4,32) \).
03

- Check for Unique Outputs

Check each output value to see if it corresponds to exactly one input value. Here, 2 corresponds to the input 1, 8 corresponds to the input 2, 18 corresponds to the input 3, and 32 corresponds to the input 4.
04

- Determine One-to-One Status

Since all output values are unique and correspond to unique input values, 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.

Function Properties
A foundational concept in mathematics is the function.
Functions assign each input a specific output. This relationship is crucial for understanding many areas of math.
Functions have various properties, one of the most important being whether they are one-to-one.
A function is considered one-to-one (or injective) if every input produces a unique output.
This means no two different inputs can produce the same output.
In simpler terms, if you change the input, the output must also change.
Unique Outputs
When checking if a function is one-to-one, focus on the uniqueness of its outputs.
Each output must pair with only one input.
For instance, in the given pairs \( (1,2), (2,8), (3,18), (4,32) \), notice that each output (2, 8, 18, 32) corresponds to a distinct and unique input.
If we had any repeated outputs paired with different inputs, the function would not be one-to-one.
Unique outputs ensure that the function is injective.
Input-Output Pairs
Understanding input-output pairs is key to analyzing functions.
Each pair is written in the form \( (x, y) \), where x is the input and y is the output.
For our function, the pairs provided are \( (1,2), (2,8), (3,18), (4,32) \).
We can see that different x-inputs (1, 2, 3, 4) all have different y-outputs (2, 8, 18, 32).
This is a necessary condition for a function to be considered one-to-one.
Reviewing these pairs step-by-step helps confirm the properties of the function.
Algebra
Algebra helps in analyzing and understanding functions.
It provides the tools to define, manipulate, and test functions.
A typical test for a one-to-one function involves ensuring that the function passes the 'Horizontal Line Test'.
If no horizontal line intersects the function's graph more than once, the function is one-to-one.
In our case, observing the input-output pairs helps us visually and numerically confirm that each input maps to a unique output.
Understanding these algebraic rules is crucial for analyzing whether functions are one-to-one.

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