Problem 85 - College Algebra and Calculus: An Applied Approach

Question

Determine whether the statements use the word function in ways that are mathematically correct. Explain your reasoning. (a) The amount in your savings account is a function of your salary. (b) The speed at which a free-falling baseball strikes the ground is a function of its initial height.

Step-by-Step Solution

Answer

Both statements (a) 'The amount in your savings account is a function of your salary' and (b) 'The speed at which a free-falling baseball strikes the ground is a function of its initial height' use the word function correctly in a mathematical sense.

Step 1: Analyze Statement (a)

Statement (a) is 'The amount in your savings account is a function of your salary'. This is mathematically correct as the amount in your savings account directly depends on your salary. For each salary, there is a corresponding amount in your savings account which makes it a function.

Step 2: Analyze Statement (b)

Statement (b) is 'The speed at which a free-falling baseball strikes the ground is a function of its initial height'. The speed at which the baseball hits the ground depends on its initial height under the influence of gravity, with no air resistance considered. Hence, for a given height, there can be a particular speed at which the ball hits the ground, making this a function and therefore the statement is mathematically correct.

Key Concepts

Understanding Dependent Variables

In the realm of functions, a dependent variable is essentially what you measure or what you focus on as the result or outcome of a function. It shifts when the independent variable changes. Let's consider the previous example of salary and savings. Here, your savings account balance changes based on your salary. Thus, the savings account amount is the dependent variable.
The dependent variable responds to the independent variable. This is key for understanding how functions work. Remember:

• Dependent variables depend on other factors.
• In an equation like \( y = f(x) \), \( y \) is often the dependent variable.
• Changes in \( x \) cause changes in \( y \).

Learning how to identify dependent variables can help you anticipate outcomes in various situations. It's vital to know what you're trying to measure or predict.

Exploring Independent Variables

Unlike dependent variables, independent variables are what you, or the situation, can change. They are "independent" because they are not affected by the variables you are measuring. In the context of functions, independent variables are the ones that drive changes in the dependent variables.
For instance, if we revisit our savings account example, the salary acts as the independent variable. Your salary can vary (higher or lower), and this variation causes changes in your account balance. Here are a few points to consider:

• The independent variable is the element you control or alter.
• In the function \( y = f(x) \), \( x \) is usually the independent variable.
• Changes in the independent variable can cause direct and predictable changes in the dependent variable.

Mastering this concept helps in both understanding problems and solving them effectively.

Real-World Applications of Functions

Functions are not mere abstract mathematical concepts; they have practical applications everywhere in the real world. They describe how one quantity changes with respect to another. This is crucial in fields like economics, physics, and other sciences.
Consider the example where the speed of a falling baseball is determined by its initial height. This serves as a simple illustration of functions in physics—describing motion and predicting outcomes based on initial conditions. Here are some more applications of functions:

• In economics, functions can represent the relationship between supply and demand.
• In biology, they can model growth rates of populations.
• In technology, they play a vital role in programming and algorithms.

Understanding these applications helps us model real-world phenomena and make informed decisions based on projected outcomes.