Question

Applications. Writing: The word "function" is used in everyday speech, such as, "The rate of advance in cancer research is a function of the amount of money invested." Find other examples such as this, and relate their meaning to the mathematical meaning of function that we have given here.

Short answer

In everyday language, 'function' often indicates a dependent relationship, similar to mathematical functions where each input has exactly one output. For example, job performance can be described as a function of hours of sleep, mathematically noted as P(h), where P is performance and h is hours of sleep.

Step-by-step solution

Understanding Mathematical Function

Understand the mathematical definition of a function. A function is a relation between a set of inputs and a set of permissible outputs where each input is related to exactly one output. For example, if we have a function f(x), for every value of x, there is exactly one corresponding value of f(x).

Identifying Functions in Everyday Speech

Identify and think about common phrases where the term 'function' is used. In everyday speech, we often say that something is 'a function of' something else when there is a dependent relationship where one thing varies according to the change in another.

Relating Everyday Examples to Mathematical Functions

Relate the examples from Step 2 to the mathematical concept of a function. For instance, the success of a business can be said to be a function of customer satisfaction. Mathematically, this could be represented by a function where the amount of success, S(x), is the output, and customer satisfaction level, x, is the input, indicating that for each level of customer satisfaction, there is a corresponding level of business success.

Writing Examples

Write out the examples found in Step 2 and express their relationship in the form of a function. Example: The amount of traffic on a road (T) can be a function of the time of day (t), expressed as T(t). At different times, t, there will be corresponding levels of traffic, T.

Key concepts

Function Applications

Functions are not just abstract mathematical concepts; they are vital tools applied in various real-world contexts. Think of a function as a machine where you input something, and you get a particular output every time. For instance, when baking a cake, the softness of the cake (output) is a function of the baking time and temperature (inputs). If we know the right amount of time and the temperature, we can predict how soft the cake will be.

To make this more concrete, let’s consider another example with environmental science. The quality of air in a city (output) can be described as a function of the amount of pollutants emitted (input). If we have a mathematical model or function for this, we could predict air quality based on emissions data, which can be incredibly useful for decision-making in public health and city planning.

The key is to observe how changing one element (the input) consistently results in a change in another (the output), and it's this consistent relationship that we harness in 'function applications' across various fields like economics, science, and engineering.

Relation Between Variables

When we talk about the relation between variables in functions, we're looking at how an alteration in one variable affects another. This is fundamental in understanding cause-and-effect relationships in real life. Take, for example, a grocery store: the total cost (output) is a relation of the number of items purchased (input). More items mean more cost, fewer items mean less cost; the cost varies with the quantity.

Economic Relationships

In economics, this concept is prevalent. The demand for a product (output) often functions in relation to its price (input). As price goes up, demand might go down, and vice versa; illustrating a relation where the two variables are inversely related. This concept helps businesses set pricing strategies that optimize profits.

Health and Medicine

In health and medicine, body mass index (BMI) is a function of a person's weight and height. Here we see a direct relationship: as weight increases with the same height, BMI increases as well. This relationship helps healthcare professionals quickly assess whether a patient's weight is in a healthy range for their height.

Input-Output Relationships

Understanding the input-output relationship is like analyzing how a change in one quantity leads to a change in another. In mathematics, this is represented by a function: you input a number (x) and get an output (f(x)). A familiar example could be a taxi service where the fare charged (output) is calculated as a function of the distance traveled (input).

Education and Grades

In education, a student's grade (output) can be considered a function of the hours spent studying (input). Although it’s not always a direct correlation since other factors come into play, there is generally an understanding that more study time can improve grades.

Technology and Computing

Computing offers a more technical example. The performance of a computer (output) could be a function of its processor speed, available memory, and software efficiency (inputs). Here, a balanced combination of these inputs is crucial for optimal output, illustrating how multiple variables can simultaneously impact the function's output. Understanding these input-output relationships empowers us to make predictions, improve systems, and optimize outcomes in various aspects of life and technology.