Problem 76 - College Algebra with Modeling and Visualization

Question

Heights A relation takes a student's height rounded to the nearest inch as input and outputs the student's name with that height. Does this relation typically compute a function? Explain.

Step-by-Step Solution

Answer

The relation does not compute a function, as a single height can correspond to multiple students.

Step 1: Understanding Relations and Functions

A relation from one set to another set is a collection of ordered pairs, where each first component (input) can be related to multiple second components (outputs). A function is a special kind of relation where each input is related to exactly one output.

Step 2: Identifying Inputs and Outputs

In this exercise, the relation takes a student's height (rounded to the nearest inch) as the input and outputs the student's name. The main question is whether a single height can correspond to only one student's name.

Step 3: Analyzing the Condition of a Function

For the relation to be considered a function, each possible height value must correspond to exactly one student name. If more than one student can have the same height value, it ceases to be a function.

Step 4: Applying the Concept

In typical classroom settings, it's common for multiple students to have the same height when rounded to the nearest inch. Therefore, a single height is likely to be paired with more than one student's name, meaning it does not fit the criteria of a function.

Step 5: Conclusion

Since there can be multiple students with the same height, the relation does not define a function, as one input (height) maps to multiple outputs (student names).

Key Concepts

Ordered Pairs

In mathematics, an ordered pair is simply a pair of elements where the order of the elements is significant. Typically, ordered pairs are written in the form \((a, b)\)where "a" represents the first component and "b" represents the second component.
In our exercise, the ordered pair would consist of a student's height as the first component and the student's name as the second component.
From this setup:

• The first value (input) is the height.
• The second value (output) is the student's name.

Ordered pairs are crucial in defining relations and functions because they clearly express how two distinct types of data are linked together.
In this case, they depict the potential interaction between student heights and names.

Inputs and Outputs

Every mathematical function or relation involves inputs and outputs. Inputs are the initial data or values fed into the system, and outputs are what you receive after the processing of these inputs. In the context of the provided exercise, inputs mean the heights of students (rounded to the nearest inch).
Outputs refer to the corresponding names of the students associated with each height.
This particular exercise is prompting us to consider whether each input (a distinct student height) returns just a single output (one specific student's name).

• If each height only maps to one name, it could be considered a function.
• If a single height maps to multiple names, it remains a relation but isn't a function.

Thinking about inputs and outputs in this organized manner helps clarify whether the system in question adheres to function criteria.

Function Criteria

The distinction between a general relation and a function lies mainly in the unique correspondence criterion.
This criterion dictates that each input of a function must be linked to exactly one output. To determine if a relation meets the function criteria, consider the following:

• Each unique input (height) should have a unique output (student name).
• If two students share the same height, the relation breaks function criteria.

In classroom scenarios, it is quite typical for several students to share the same height, especially when we round to the nearest inch.
This overlapping would mean a single height as input could result in several student names as outputs, breaching the single-output rule of functions. Because this condition is often unmet in such real-world examples, the described relation doesn’t satisfy function criteria, making it a non-function.