Exemplo 4 de Algebra and Trigonometry, 3.4 Composition of Functions

Investigating the Order of Function Composition

Problema e solução em inglês, como no livro original.

Suppose $f (x)$ gives miles that can be driven in $x$ hours and $g (y)$ gives the gallons of gas used in driving $y$ miles. Which of these expressions is meaningful: $f (g (y))$ or $g (f (x)) ?$

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Solução
The function $y = f (x)$ is a function whose output is the number of miles driven corresponding to the number of hours driven.

$number of miles = f (number of hours)$
The function $g (y)$ is a function whose output is the number of gallons used corresponding to the number of miles driven. This means:

$number of gallons = g (number of miles)$
The expression $g (y)$ takes miles as the input and a number of gallons as the output. The function $f (x)$ requires a number of hours as the input. Trying to input a number of gallons does not make sense. The expression $f (g (y))$ is meaningless.

The expression $f (x)$ takes hours as input and a number of miles driven as the output. The function $g (y)$ requires a number of miles as the input. Using $f (x)$ (miles driven) as an input value for $g (y),$ where gallons of gas depends on miles driven, does make sense. The expression $g (f (x))$ makes sense, and will yield the number of gallons of gas used, $g,$ driving a certain number of miles, $f (x),$ in $x$ hours.

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The gravitational force on a planet a distance r from the sun is given by the function $G (r) .$ The acceleration of a planet subjected to any force $F$ is given by the function $a (F) .$ Form a meaningful composition of these two functions, and explain what it means.

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