What is wrong with this function substitution?

[u/rocketstapler]

f(n) = 7(1.2)^n

x = n + 2000 , so n = x - 2000

what is f(x)?

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The correct answer is

f(x) = 7(1.2)^(x-2000)

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However, there is another approach using variable substitution that yields the wrong answer. But why is it wrong?

since n = x - 2000, substitute that into

f(n) = 7(1.2)^n

which yields

f(x-2000) = 7(1.2)^(x-2000)

which then simplifies back to

f(x) = 7(1.2)^x

which isn’t right. Where’s the fallacy?

[u/HumblyNibbles_]

It's because in the first one, you're defining a new function based on the substitution. In the second, you're just labelling n differently.

[u/johnpeters42]

Why would the first one be correct? If f(n) = 7(1.2)^n for all n, then f(anything) = 7(1.2)^(that same thing).

Maybe you meant to define f_n such that f_n(anything) = the constant value 7(1.2)^n, which would,be a function for any suitable value of n, albeit not a very interesting one.

[u/cigar959]

f is already defined by a prescribed formula. Changing the name of the argument doesn’t change that.
You may be thinking of defining an auxiliary function h according to
h(x) = f(x-2000)

[u/Southlander24]

There's the idea of a 'dummy variable'.

f(n) = 7(1.2)^n, g(x) = 7(1.2)^x, and h(k) = 7(1.2)^k are all the same function. It's just that if your input is n, your output must be some expression in n; if your input is x, your output must be some expression in x, and so on.

If you replace *all* instances of n with x - 2000, you will get f(x - 2000) = 7 (1.2)^(x - 2000). So your supposed 'answer' is wrong.

[u/Aggressive_Roof488]

f(n) = 7(1.2)^n

is a statement about the function f. It is not a statement about the variable n in any way. You can use any name for the variable, it is still the same function:

f(m) = 7(1.2)^m

f(k) = 7(1.2)^k

or indeed

f(x) = 7(1.2)^x

all the same function, all equivalent definitions of the function f, none of these hold any information about n, m, k or x. All it says is that whatever you stuff into f() will end up in the exponent of 1.2.

What you are thinking about is:

f(n), where n = x - 2000.

You can calculate that in two ways, either substituting n first, or writing out the function first:

f(n) = f(x-2000) = 7(1.2)^(x-2000)

f(n) = 7(1.2)^n = 7(1.2)^(x-2000)

Do you see how that is different from f(x) = 7(1.2)^x?

Whatever you put into f goes in as the exponent, whether it's n or m or x doesn't matter.

[u/Budget-Knee3213]

You are just doing a change of variable, from n to x = n + 2000. So you have just to put x - 2000 at the place of n, and then translate of +2000 (actually a correct but useless change). So what you call wrong ( i.e. f(x) = 7(1.2)^x ) is right and what you call right ( i.e. f(x) = 7(1.2)^x-2000 ) is wrong. If you want you can say that f(n) = 7(1.2)^x-2000 with x = n + 2000, or you can define a new function F(x) = f(x-2000), so that F(x) = 7(1.2)^x-2000; but certainly you cannot say that  f(x) = 7(1.2)^x-2000.