Question

as with function formulas, there are advantages to using function notation to define a composite - function. lets explore function composition notation and use this notation to represent quantities values.
one goal is to use function notation to represent the total fuel cost of a trip in terms of the distance (in miles) of the trip. in doing so we first need to define a function, f, to determine the volume (in gallons) of gasoline used, v, in terms of the distance (in miles) of a trip, d.
a. represent v using function notation.
v = f(d) preview
b. define a function f that determines the volume of gasoline (in gallons) used, v = f(d), in terms of the distance (in miles) of a trip, d.
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c. define a function g that determines the fuel cost, c = g(v), in terms of the volume (in gallons) of gasoline used on the trip, v.
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d. define a function g that determines the fuel cost, c, in terms of d, by substituting f(d) for v in your previous definition.
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e. represent the total fuel cost of a 267 - mile trip using function notation.
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f. represent how much greater the total fuel cost of a 470 - mile trip is than the total fuel cost of a 350 - mile trip using function notation.
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question 4 points possible: 5 unlimited attempts.

Explanation:

Step1: Represent volume using function notation

The volume $v$ in terms of distance $d$ is given as $v = f(d)$ which is already provided correctly in the problem - statement for part a.

Step2: Define function $f$ for volume

If we assume the car has a fuel - efficiency of $m$ miles per gallon (mpg), then the volume of gasoline used $v$ in terms of distance $d$ is $f(d)=\frac{d}{m}$, where $m$ is the number of miles the car can travel per gallon of gasoline.

Step3: Define function $g$ for fuel cost

If the cost of gasoline is $c$ dollars per gallon, then the function $g$ that determines the fuel cost $c$ in terms of the volume $v$ of gasoline used is $g(v)=cv$.

Step4: Define function $g$ in terms of $d$

Substitute $v = f(d)=\frac{d}{m}$ into $g(v)$. We get $g(f(d))=c\times\frac{d}{m}=\frac{cd}{m}$.

Step5: Represent total fuel cost for 267 - mile trip

We use the composition $g(f(d))$. For $d = 267$, the total fuel cost is $g(f(267))$.

Step6: Represent difference in fuel costs

The total fuel cost of a 470 - mile trip is $g(f(470))$ and of a 350 - mile trip is $g(f(350))$. The difference is $g(f(470))-g(f(350))$.

Answer:

a. $v = f(d)$
b. $f(d)=\frac{d}{m}$ (assuming fuel - efficiency $m$ mpg)
c. $g(v)=cv$ (assuming cost per gallon $c$)
d. $g(f(d))=\frac{cd}{m}$
e. $g(f(267))$
f. $g(f(470)) - g(f(350))$