Question

A relation expressed verbally is given. (a) What is the domain and the range of the relation? (b) Express the relation using a mapping. (c) Express the relation as a set of ordered pairs. The density of a gas under constant pressure depends on temperature. Holding pressure constant at 14.5 pounds per square inch, a chemist measures the density of an oxygen sample at temperatures of \(0,22,40,70,\) and \(100^{\circ} \mathrm{C}\) and obtains densities of \(1.411,1.305,1.229,1.121,\) and \(1.031 \mathrm{~kg} / \mathrm{m}^{3},\) respectively.

Short answer

Domain: \(\{0^{\text{c}}, 22^{\text{c}}, 40^{\text{c}}, 70^{\text{c}}, 100^{\text{c}}\}\). Range: \(\{1.411, 1.305, 1.229, 1.121, 1.031 \text{ } \text{kg/m}^{3}\}\). Ordered Pairs: \((0^{\text{c}}, 1.411 \text{ kg/m}^{3}), (22^{\text{c}}, 1.305 \text{ kg/m}^{3}), (40^{\text{c}}, 1.229 \text{ kg/m}^{3}), (70^{\text{c}}, 1.121 \text{ kg/m}^{3}), (100^{\text{c}}, 1.031 \text{ kg/m}^{3})\).

Step-by-step solution

- Identify the Given Values

Identify the values given in the problem. The temperatures are: \(0^{\text{c}}, 22^{\text{c}}, 40^{\text{c}}, 70^{\text{c}}, 100^{\text{c}}\). The corresponding densities are: \(1.411, 1.305, 1.229, 1.121, 1.031 \text { } \text{kg/m}^{3}\).

- Determine the Domain

The domain consists of all the input values (temperatures). Hence, the domain is \[ \text{Domain} = \{0^{\text{c}}, 22^{\text{c}}, 40^{\text{c}}, 70^{\text{c}}, 100^{\text{c}} \} \].

- Determine the Range

The range consists of all the output values (densities). Hence, the range is \[ \text{Range} = \{ 1.411, 1.305, 1.229, 1.121, 1.031 \text { } \text{kg/m}^{3} \} \].

- Express the Relation Using a Mapping

Using a mapping diagram, match each input with its corresponding output. This relation maps: \[ 0^{\text{c}} \rightarrow 1.411 \text{ kg/m}^{3}, 22^{\text{c}} \rightarrow 1.305 \text{ kg/m}^{3}, 40^{\text{c}} \rightarrow 1.229 \text{ kg/m}^{3}, 70^{\text{c}} \rightarrow 1.121 \text{ kg/m}^{3}, 100^{\text{c}} \rightarrow 1.031 \text{ kg/m}^{3} \].

- Express the Relation as a Set of Ordered Pairs

Each input corresponds to an output, forming ordered pairs. Hence, the set of ordered pairs is \[ \{ (0^{\text{c}}, 1.411 \text{ kg/m}^{3}), (22^{\text{c}}, 1.305 \text{ kg/m}^{3}), (40^{\text{c}}, 1.229 \text{ kg/m}^{3}), (70^{\text{c}}, 1.121 \text{ kg/m}^{3}), (100^{\text{c}}, 1.031 \text{ kg/m}^{3}) \} \].

Key concepts

Domain and Range

In mathematics, the concept of 'domain and range' is fundamental to understanding functions and relations.
The domain of a relation or function is the complete set of all possible input values (also known as independent variables).
These are the values you can plug into the function.

For example, in the given exercise, the domain consists of the temperatures measured at different points, specifically: \[ \text{Domain} = \{0^{\text{c}}, 22^{\text{c}}, 40^{\text{c}}, 70^{\text{c}}, 100^{\text{c}} \} \]

On the other hand, the range refers to the complete set of all possible output values (dependent variables).
These are the values you get out after performing the function on the domain.
In our example, the range includes the densities of oxygen at those temperatures:
The range is thus:
\[ \text{Range} = \{1.411, 1.305, 1.229, 1.121, 1.031 \text { } \text{kg/m}^{3} \} \]

Mapping Relations

Mapping relations visually show the connection between elements of the domain and elements of the range.
Think of it as drawing arrows from each input to its corresponding output.
In our problem, we used a mapping diagram where every temperature (domain) maps to its corresponding density (range).

To express this relation using a mapping, we clearly pair each temperature to a specific density.
It can be represented as follows:
\ \
- \(0^{\text{c}} \rightarrow 1.411 \text{ kg/m}^{3}\)
- \(22^{\text{c}} \rightarrow 1.305 \text{ kg/m}^{3}\)
- \(40^{\text{c}} \rightarrow 1.229 \text{ kg/m}^{3}\)
- \(70^{\text{c}} \rightarrow 1.121 \text{ kg/m}^{3}\)
- \(100^{\text{c}} \rightarrow 1.031 \text{ kg/m}^{3}\)

This type of diagram is useful for visually understanding how each input is connected to an output.

Ordered Pairs

Ordered pairs are another way to express relations.
An ordered pair is written in the form \((input, output)\), showing the relationship between two quantities.

In algebra, ordered pairs are expressed as \((x, y)\), where \(x\) is the input value (from the domain) and \(y\) is the output value (from the range).
For the given exercise, the ordered pairs are:
\[ \{ (0^{\text{c}}, 1.411 \text{ kg/m}^{3}), (22^{\text{c}}, 1.305 \text{ kg/m}^{3}), (40^{\text{c}}, 1.229 \text{ kg/m}^{3}), (70^{\text{c}}, 1.121 \text{ kg/m}^{3}), (100^{\text{c}}, 1.031 \text{ kg/m}^{3}) \} \]
These pairs clearly show each temperature corresponding to its measured density.
Ordered pairs are crucial in graphing relations and understanding functions in coordinate geometry.

Temperature and Density

Understanding the relation between temperature and density is important in various scientific fields, especially chemistry and physics.
In the given exercise, we see how the density of oxygen changes as the temperature varies while keeping the pressure constant at 14.5 pounds per square inch.

As temperatures increase, the corresponding densities decrease.
This is typical for gases; as temperature rises, the molecules move more rapidly and spread apart, reducing density.
Hence, the measured densities decrease from \(1.411 \text{ kg/m}^{3}\) at \(0^{\text{c}}\) to \(1.031 \text{ kg/m}^{3}\) at \(100^{\text{c}}\).

In summary, the relationship highlights an inverse proportionality between temperature and density for gases.