Question

Find a formula for \(g\) by scaling the input and/or output of \(f\). Let \(f(w)\) give the expected weight in lbs of an infant at one month's age if its weight at birth is \(w\) lbs, and \(g(m)\) give the expected weight in \(\mathrm{kg}\) if the weight at birth is \(m \mathrm{~kg}\). Use the fact that \(1 \mathrm{lb}\) equals \(0.454 \mathrm{~kg}\).

Short answer

Answer: The formula for \(g(m)\) in terms of \(f(w)\) is \(g(m) = 0.454 f\left(\frac{m}{0.454}\right)\).

Step-by-step solution

Convert the input/output between lbs and kg using the conversion factor

Given the conversion factor, we can convert lbs to kg and vice versa using the following relations: To convert \(w\) lbs to \(m\) kg: \(m = 0.454w\) To convert \(m\) kg to \(w\) lbs: \(w = \frac{m}{0.454}\) We want to find a relation between \(g(m)\) and \(f(w)\), so we can first convert \(m\) to \(w\) using the second relation.

Write the formula for \(g(m)\) in terms of \(f(w)\)

Now we can express the input of function \(f(w)\) in terms of \(g(m)\): \(w = \frac{m}{0.454}\) Substitute this expression into \(f(w)\): \(g(m) = f\left(\frac{m}{0.454}\right)\) However, we need to convert the output of \(f\) to kg. Since \(f\) gives the expected weight in lbs, we can convert the output by multiplying it by the conversion factor: \(g(m) = 0.454 f\left(\frac{m}{0.454}\right)\) This is the formula for \(g(m)\) in terms of \(f(w)\).

Key concepts

Weight Conversion

When dealing with weight conversion, it's important to comprehend the process of changing one unit of measurement to another. In our given problem, we are shifting between pounds (lbs) and kilograms (kg). The fundamental conversion factor you need to remember is that 1 lb is equal to 0.454 kg.

Equipped with this conversion factor, you can smoothly switch measurements:

• To convert pounds to kilograms: Multiply the number of pounds by 0.454. For example, turning 10 lbs into kg equals 10 lbs \( \times \) 0.454, which results in 4.54 kg.
• Conversely, to convert kilograms to pounds: Divide the number of kilograms by 0.454. This would mean turning 2 kg into lbs gives you \( \frac{2}{0.454} \approx 4.41 \) lbs.

This simple multiplication or division allows for seamless switching between these standard weight measurements, essential when comparing outcomes in differing unit contexts.

Function Composition

Function composition is like building a new function by integrating two existing functions. Considering our problem, we have two functions: one describing weight be it in lbs, denoted as \( f(w) \), and another in kg, denoted as \( g(m) \). Our task is to create a bridge between these two based on the unit conversion.

The idea is to turn the function for lbs into a function for kg. Begin by performing a unit conversion on the input, which is originally in kg for the function \( g(m) \), to transform it into lbs before applying function \( f \). Once you have calculated the expected weight using \( f \), you'll receive results in lbs. Then, re-convert these results back to kg using the known conversion factor.

• The equation \( w = \frac{m}{0.454} \) changes the input \( m \) kg to \( w \) lbs.
• Post input conversion, applying the function \( f(w) \) gives the weight outcome in lbs.
• Finally, return the output from lbs to kg, concluding with \( g(m) = 0.454 f\left(\frac{m}{0.454}\right) \).

Function composition allows the above transformation to deliver meaningful results across the different measurement units.

Input Scaling

Input scaling involves modifying the arguments of a function to reflect changes in the physical or assumed context. Here, we aim to adjust the input to accommodate unit shifts from kg to lbs, effectively scaling the input.

By working through scaling, the primary goal is to ensure the function accurately describes the new scenario despite differing initial values. With weight units, this is accomplished by scaling down or up depending on the direction of conversion.
Take the original input \( m \) in kg and scale it to function \( f \) which expects lbs:

• Use \( w = \frac{m}{0.454} \) to scale the input from kg to lbs, matching the required input dimensionality for \( f \).
• Apply function \( f \), bearing in mind this adjustment, yields outputs that initially relate to the input weight measurement (lbs) but need reconversion to kg.
• This input scaling is further reinforced by verifying consistency across both input and output scales, converging back to simplified results \( g(m) = 0.454 f\left(\frac{m}{0.454}\right) \).

Successful input scaling harmonizes the function with new operational parameters, essential for adapting solutions to different input environments.