91Ó°ÊÓ

Answer the question and then write a short explanation. The table shows the number of hazardous waste sites on the National Priority List for the Superfund program in 2008 for seven states in the top ten. If this correspondence is considered to be a function that pairs each state with its number of hazardous waste sites, is it one-to-one? If not, explain why. $$\begin{array}{|c|c|}\hline \text { State } & \text { Number of Sites } \\\\\hline \text { New Jersey } & 116 \\\\\hline \text { California } & 97 \\\\\hline \text { Pennsylvania } & 96 \\\\\hline \text { New York } & 86 \\\\\hline \text { Florida } & 52 \\\\\hline \text { Illinois } & 49 \\\\\hline \text { Texas } & 49 \\\\\hline\end{array}$$

Step by step solution

01

Define One-to-One Function

A function is considered one-to-one if every element of the range (output) is paired with exactly one element of the domain (input). This means no two different inputs can map to the same output.

02

Identify Inputs and Outputs

In this problem, the inputs (domain) are the states, and the outputs (range) are the number of hazardous waste sites in those states.

03

Examine the Table Data

The table given shows the following pairs:- New Jersey: 116- California: 97- Pennsylvania: 96- New York: 86- Florida: 52- Illinois: 49- Texas: 49

04

Check for Repeated Outputs

Examine the outputs to see if any number is repeated. Here, the number 49 appears twice (for Illinois and Texas).

05

Determine if One-to-One

Since the number 49 is paired with two different states (Illinois and Texas), the function is not one-to-one. A one-to-one function requires unique outputs for each input.

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Key Concepts

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

Function Definition

A function is a special relationship between two sets: a domain (inputs) and a range (outputs). In a function, each input is paired with exactly one output.
For example, if you think of a vending machine, selecting a button (input) gives you a specific snack (output). The key point is that every button corresponds to one unique snack.
The machine should not give two different snacks for the same button.

Domain and Range

The domain of a function is the complete set of possible inputs, while the range consists of all the possible outputs.
In this exercise, the domain is the seven states: New Jersey, California, Pennsylvania, New York, Florida, Illinois, and Texas.
The range is the number of hazardous waste sites in these states, which are the numbers: 116, 97, 96, 86, 52, and 49.

• Domain: {New Jersey, California, Pennsylvania, New York, Florida, Illinois, Texas}
• Range: {116, 97, 96, 86, 52, 49}

Repeated Outputs

For a function to be one-to-one, each output in the range must be unique, meaning no two inputs must share the same output.
In our table, the number 49 appears twice, once for Illinois and once for Texas. This repetition indicates that the function is not one-to-one since these two states (inputs) have the same number of sites (output).
Remember: One-to-one functions ensure that every single input maps to a different output.

Unique Pairing

A one-to-one function demands a unique pairing between inputs and outputs. This way, different inputs cannot produce the same output.
Let's revisit our example: In the table of states, Illinois and Texas have the same output (49 hazardous sites). Since different inputs (states) share the same output (number of sites), unique pairing does not occur.
For a function to be truly one-to-one, both the domain and range must have a perfect one-to-one correspondence.