SOLUTION: 22. (See Week 5 Lecture page �Composite Functions� for a starting point.) The population of viruses in an influenza culture after t hours is given by the function x(t) = 4e^t/2.5

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Question 1090122: 22. (See Week 5 Lecture page �Composite Functions� for a starting point.)
The population of viruses in an influenza culture after t hours is given by the function
x(t) = 4e^t/2.5
The cost y in dollars for a new automated microscope to count x viruses in a sample is
y(x) = ln 2x
Create the composite function that calculates the cost y in dollars of counting the number of viruses in an influential culture after t hours.
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
composite function is a function that operates on another function as its argument.

example:

f(x) = x^2

g(x) = 3x

g(f(x)) = 3 * x^2

the function f(x) replaces the argument of x in g(x).

you have x(t) = 4e^(t/2.5)

you have y(x) = ln(2x)

the function y(x(t)) replaces the argument of x in y(x) with the output of 4e^(t/2.5) in g(t)

this goes to the heart of the definition of a function.

in the function x(t) = 4e^(t/2.5):

x(t) is the name of the function.
t is the input of the function.
4e^(t/2.5) is the output of the function.

here's what i found on the web that looks like a pretty good reference.
read it all the way through because it describes how to replace the original argument with another expression when evaluating that expression within the function.

https://mathbitsnotebook.com/Algebra1/Functions/FNNotationEvaluation.html

here's also a reference on composite functions i found on the web.

http://www.mesacc.edu/~scotz47781/mat120/notes/composition/composite_functions_intro.pdf

i followed the rules in solving your problem.

the argument in the first function was replaced with the output of the second function.

once again, for ease of reference.

the first function is x(t) = 4e^(t/2.5)

the argument of this function is t and the output of this function is 4e^(t/2.5).

this tells you the population of viruses after t hours.

the second function is y(x) = ln(2x)

the argument of this function is x and the output of this function is ln(2x).

don't confuse the function name of x(t) with the argument of x in the function name of y(x).

they're two different things.

you could have named the function x(t) anything else and the function would have been the same.

some examples:

f(t) = 4e^(t/2.5)
g(t) = 4e^(t/2.5)
h(t) = 4e^(t/2.5)
x(t) = 4e^(t/2.5)

they all point to the same relationship between the argument of t and the output of e^(t/2.5)

your other function of y(x) = ln(2x) could also have many names.

n(x) = ln(2x)
p(x) = ln(2x)
q(x) = ln(2x)
y(x) = ln(2x)

all of these functions have an argument of x and an output of ln(2x).

back to your original problem.

your functions are:

x(t) = 4e^(t/2.5)

y(x) = ln(2x)

note the parentheses.
they make it much clearer as to where each expression within the overall expression belong.
hopefully i got it right.

x(t) represents the number of viruses after t hours
y(x) represents the cost of a new microphone to count the number of viruses.

therefore y(x(t)) represents the cost of a new microphone to count the number of viruses after t hours.

the argument of y(x) is replaced by the output of the function x(t).

x(t) = 4e^(t/2.5)

y(x) = ln(2x)

y(x(t)) = ln(2 * 4e^(t/2.5))

i believe you can simplify that to y(x(t)) = ln(8e^(t/2.5))

that's what i get.

let me know if you have any further questions regarding this.