Question

Find a formula for \(w\) by scaling the input of \(f\). Let \(f(n)\) give the average time in seconds required for a computer to process \(n\) megabytes (MB) of data, and \(w(r)\) the time required for \(r\) gigabytes (GB). Use the fact that \(1 \mathrm{~GB}\) equals \(1024 \mathrm{MB}\).

Short answer

Answer: The formula for \(w(r)\) is \(w(r) = f(1024r)\).

Step-by-step solution

Express the given relation between gigabytes and megabytes in terms of variables

We know that \(1 \mathrm{~GB} = 1024 \mathrm{~MB}\). Let's say we have \(r\) gigabytes of data. Then, the equivalent amount of data in megabytes is given by \(n = 1024r\).

Substitute the relation derived in Step 1 into the function \(f(n)\)

Now that we have expressed the relationship between \(r\) and \(n\), we can substitute this into the given function \(f(n)\). Instead of \(f(n)\), we will use \(f(1024r)\), since \(n = 1024r\).

Define the function \(w(r)\)

Finally, let's define the function \(w(r)\) as the time required for \(r\) gigabytes of data, which is equal to \(f(1024r)\). So, the formula for \(w(r)\) is: \[w(r) = f(1024r)\] This formula connects the time required to process \(r\) gigabytes of data, \(w(r)\), to the time required to process \(n\) megabytes of data, \(f(n)\).

Key concepts

Scaling Functions

Understanding how to scale functions is an essential skill, especially when dealing with different measurement units or data types. Scaling simply means adjusting the function to fit the new data dimensions while keeping its behavior consistent. In the context of the original exercise, we're scaling the function to transition from megabytes (MB) to gigabytes (GB).

This is achieved by a straightforward unit conversion where 1 GB equals 1024 MB. By realizing this conversion factor, we can transform the internal variable of the function. So, if our original function is based on megabytes, to adapt it for gigabytes, we scale the input by multiplying with 1024. This transforms the function from processing data measured in MB, to that measured in GB.

Scaling not only helps in converting the unit or scale of inputs but also maintains the integrity of the relationship among the variables in different contexts. It is a powerful tool to ensure consistent applicability of functions regardless of their input size or unit.

Data Conversion

Data conversion in the context of functions is about transforming data from one form or unit to another, allowing for meaningful comparisons and calculations. It plays a vital role in scenarios where inputs are in varied forms, like the conversion from gigabytes to megabytes seen in this exercise.

When we convert gigabytes to megabytes, we're essentially preparing the data to be processed by a particular function designed for a specific unit. This involves understanding the conversion factor between these units, with 1 GB equating to 1024 MB. By applying this factor, we multiply the gigabyte value by 1024 to convert it into megabytes before feeding it into the function. This ensures that the function \(f(n)\) can handle the data correctly, as it was initially intended to process inputs in megabytes.

Proper data conversion is crucial in computational operations, ensuring that comparisons and calculations remain accurate, regardless of the initial data presentation.

Function Notation

Function notation is an essential concept in mathematics, providing a succinct way to define and handle functions. It allows us to clearly express the dependency of one variable on another, by using symbols such as \(f(n)\) or \(w(r)\). This notation encapsulates the function's rule and its applied variable.

In the exercise, \(f(n)\) represents the original function that computes the time required for processing data in megabytes. By using function notation, it describes how one variable \(n\) affects another, in this case, the processing time. When applying a transformation, like converting MB to GB, we redefine the function with the new notation \(w(r)\). This alteration signifies the new relationship between the input in gigabytes and the output, based on the original function's rule.

Function notation is not just symbolic; it forms the foundational language that communicates mathematical concepts clearly and efficiently, offering insight into variable relationships and transformations in mathematical expressions.