Problem 1 - Algebra for College Students

Question

What does it mean to say that b is a function of a?

Step-by-Step Solution

Answer

To say that b is a function of a means that each value of a corresponds to exactly one value of b, usually expressed as b = f(a).

Step 1: Understand the Context of a Function

A function describes a special relationship between two variables. If we say that b is a function of a, we are indicating that there is a relationship where each value of a corresponds to exactly one value of b.

Step 2: Identify the Dependent and Independent Variables

In this relationship, 'a' is identified as the independent variable and 'b' as the dependent variable. This means that the value of b depends on the value of a.

Step 3: Describe the Functional Expression

The functional relationship is usually expressed in the form b = f(a), where f represents the rule or function that transforms a into b.

Step 4: Real-World Example

Consider a real-world example where a is the number of hours worked and b is the earnings. Here, b is a function of a because earnings depend on the number of hours worked.

Key Concepts

Independent Variables

In functions, the independent variable is the one you choose or control in an experiment or real-world scenario. It’s often represented by the letter 'a' in algebra. For example:

• Number of hours spent studying (you choose the number of hours)
• Amount of money invested (you decide how much to invest)

The independent variable is not affected by other variables within the function.
It is the cause, or input, in the relationship described by a function.

Dependent Variables

The dependent variable changes in response to the independent variable. It’s often represented by the letter 'b'. For example:

• Test score (dependent on hours studied)
• Interest earned (dependent on amount invested)

The dependent variable depends on the independent variable, acting as the effect or output in a function.

Functional Expressions

A functional expression describes how the independent variable transforms into the dependent variable. Usually, it takes the form b = f(a). The letter 'f' represents the function or rule that applies to 'a' to produce 'b'.

• b = 2a represents doubling the value of a to get b
• b = a^2 shows squaring a to get b

This mathematical notation precisely defines the relationship within the function.

Real-World Examples

It’s often easier to understand functions through real-world examples. Consider these practical situations:

• If a is the temperature in Celsius and b is the temperature in Fahrenheit, b = (9/5)a + 32
• If a is the distance driven and b is the amount of gas used, b might be estimated by b = 0.05a (assumes constant consumption rate)

Each example shows how one value (independent) affects another (dependent), making it clear that the dependent variable is a function of the independent variable.