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. A researcher wants to investigate how weight depends on height among adult males in Europe. She visits five regions in Europe and determines the average heights in those regions to be \(1.80,1.78,1.77,1.77,\) and 1.80 meters. The corresponding average weights are \(87.1,86.9,83.0,84.1,\) and \(86.4 \mathrm{~kg},\) respectively.

Short answer

(a) Domain: 1.77, 1.78, 1.80 meters. Range: 83.0, 84.1, 86.4, 86.9, 87.1 kg. (b) Height → Weight mapping: 1.80 → 87.1, 86.4; 1.78 → 86.9; 1.77 → 83.0, 84.1. (c) Ordered pairs: (1.80, 87.1), (1.80, 86.4), (1.78, 86.9), (1.77, 83.0), (1.77, 84.1).

Step-by-step solution

Identify Heights and Weights

List the given average heights and weights. The heights are: 1.80, 1.78, 1.77, 1.77, and 1.80 meters. The corresponding weights are: 87.1, 86.9, 83.0, 84.1, and 86.4 kg.

Determine the Domain

The domain of the relation consists of all possible heights. Since not all heights are unique, list the distinct heights: 1.77, 1.78, and 1.80 meters.

Determine the Range

The range of the relation comprises all corresponding weights. List all distinct weights: 83.0, 84.1, 86.4, 86.9, and 87.1 kg.

Mapping the Relation

Express the relation through a mapping of heights to weights: 1.80 meters → 87.1 kg, 1.80 meters → 86.4 kg,1.78 meters → 86.9 kg,1.77 meters → 83.0 kg, 1.77 meters → 84.1 kg.

Ordered Pairs

Express the mapping relation as a set of ordered pairs: (1.80, 87.1), (1.80, 86.4), (1.78, 86.9), (1.77, 83.0), and (1.77, 84.1).

Key concepts

Relation Mapping

Before diving into relation mapping, let's clarify what we mean by a relation in mathematics. A relation pairs elements from one set with elements of another set. In this case, we are pairing heights and weights.

When we talk about mapping a relation, we mean creating a visual representation that shows how elements from one set are related to elements of another set. We start with two sets: the domain (all possible heights) and the range (all corresponding weights). Each height is connected, or 'mapped', to its respective weight.

Imagine a list of heights on one side and a list of corresponding weights on the other side. Drawing arrows to connect each height to its weight creates a clear picture. For our example, the mapping looks like this:

• 1.80 meters → 87.1 kg
• 1.80 meters → 86.4 kg
• 1.78 meters → 86.9 kg
• 1.77 meters → 83.0 kg
• 1.77 meters → 84.1 kg

Even though some heights repeat, each instance is linked to a specific weight. This visual helps us see the data more clearly and understand the relationship between heights and weights better. Don't worry if you see arrows going in multiple directions from the same height, as several weights can correspond to one height.

Ordered Pairs

Ordered pairs are fundamental in understanding relations. An ordered pair consists of two elements: the first is from the domain, and the second is from the range. It's written in the form \((a, b)\), where \(a\) is the height and \(b\) is the weight in our case.

The idea behind using ordered pairs is straightforward. They give a precise way to match each height with its corresponding weight. When we write the ordered pairs for our given exercise, they look like this:

• (1.80, 87.1)
• (1.80, 86.4)
• (1.78, 86.9)
• (1.77, 83.0)
• (1.77, 84.1)

Each pair tells us exactly which height corresponds to which weight. Ordered pairs are particularly handy because they eliminate any ambiguity that might arise from repeating elements. For example, even though 1.80 meters appears twice, we know exactly which weight is related to each instance because of the ordered pair notation. Writing relations as ordered pairs, we can systematically analyze the data and work with various mathematical operations more efficiently.

Heights and Weights Relation

The relation between heights and weights in this example helps us explore real-world data through mathematical concepts. We are given the average heights and weights from five European regions. The goal is to understand how these two sets of data interrelate.

In this relation, heights act as the inputs (domain), while weights are the outputs (range). Each height has a corresponding weight. For instance, two regions with an average height of 1.80 meters each have weights of 87.1 kg and 86.4 kg, respectively. This scenario shows how one input can map to multiple outputs.

Analyzing the heights and weights in this manner allows researchers to uncover trends and patterns. If the same height frequently appears with specific weights, researchers can make educated guesses about the relationship between height and weight among the population studied. This type of relation is vital in fields like health and nutrition, where understanding how physical measurements interact can lead to better health guidelines and policies.

It's fascinating to see how combining mathematics and real-world data creates valuable insights. The exercise of mapping heights to weights and expressing them as ordered pairs shows the practical application of mathematical concepts in understanding everyday phenomena.