Problem 10 - Algebra for College Students

Question

In each situation determine whether \(y\) is a function of \(x\). Explain your answer See Example I. Consider all students taking algebra at your school. Let \(x\) represent a student's height to the nearest inch and \(y\) represent the student's IQ.

Step-by-Step Solution

Answer

No, y is not a function of x because the same height can correspond to different IQs.

Step 1: Identify Variables

In this problem, let’s identify the variables. Here, variable \(x\) represents a student's height (in inches), and variable \(y\) represents the student's IQ.

Step 2: Understand Function Definition

A function is a relation in which every input (or x-value) is associated with exactly one output (or y-value). This means each student's height should correspond to only one unique IQ.

Step 3: Examine Relationship

Evaluate whether it is possible for two students to have the same height but different IQs. Since different students can indeed have the same height but distinct IQ values, multiple y-values can correspond to the same x-value.

Step 4: Conclusion

Since one student's height can correspond to multiple IQ scores, this indicates that y is not a function of x.

Key Concepts

Function Definition

A function in algebra is a type of relationship where each input has a single, unique output. Think of it like a vending machine: you put in one coin (input), and you get one item (output). If every coin you put in gave you the same item back, then the vending machine works as a function.
Mathematically, let's denote functions as follows: If we have an input, say, variable \(x\), and an output, variable \(y\), then we can write the function as \(y=f(x)\).
The key idea here is that for each value of \(x\) (input), there is exactly one value of \(y\) (output). If you find any two same \(x\) values giving different \(y\) values, it means that it is not a function.

Independent and Dependent Variables

In algebraic functions, we talk about independent and dependent variables.
The independent variable is the input, the one we can choose freely. In our case, height \(x\) is the independent variable.
The dependent variable is the output, the one that depends on the input. Here, the IQ \(y\) is the dependent variable.
To summarize:

• The independent variable (\(x\)): the variable you control or choose
• The dependent variable (\(y\)): the result or outcome based on the independent variable

Usually, we express functions as \(y=f(x)\), indicating that \(y\) depends on \(x\).
In our example from the exercise, if every height (independent variable) corresponded to only one IQ score (dependent variable), then \(y\) would be a function of \(x\).

One-to-One Relationships

In functions, a one-to-one relationship is a special kind of function where each input is associated with exactly one unique output, and each output is associated with exactly one unique input.
This means that no two different input values share the same output value.
Here is an example:

• If height \(x\) = 60 inches maps to an IQ \(y\) = 110, and height \(x\) = 65 inches maps to IQ \(y\) = 120, this relationship is fine.
• But if height \(x\) = 60 inches maps to IQ \(y\) = 110 and also to IQ \(y\) = 130, it violates the function definition.

In our exercise, multiple students of the same height can have different IQ scores, which indicates that we do not have a one-to-one relationship.
Hence, the relationship between height and IQ does not form a function.
It's crucial to remember these distinctions when determining whether a relationship is a function.