As the title suggests, the question is rather simple:
"$f(x)$ = $e^x$ + x. Write $f(x+3)$ in terms of $f(x)$."
I encountered this problem in an precalc textbook, in the chapter regarding functions, and couldn't find the solution anywhere in the book. Although I have made progress in finding a solution (see below), I'm not quite sure if my steps are right, and whether or not my solution is in the simplest form possible.
My logic goes as follows:
$f(x+3) =e^{x+3} + x + 3$
Set f(x) = y.
$y = e^x + x$
$e^x = y - x$
$ln(e^x) = ln(y-x)$
$x = ln(y -x)$
I was able to notice that this is recursive, and got the following:
$x = ln(y - ln(y - ln(y - \dots)))$
Plugging this into our expression for $f(x+3)$, we get:
$f(x+3) = e^3(e^x) + x + 3$
$f(x+3) = e^3(e^{ln(y - ln(y - ln(y - \dots)))}) + ln(y - ln(y - ln(y - \dots))) + 3$
$f(x+3) = e^3(y - ln(y - ln(y - \dots)))) + ln(y - ln(y - ln(y - \dots))) + 3$
$f(x+3) = e^3(f(x) - ln(f(x) - ln(f(x) - \dots)))) + ln(f(x) - ln(f(x) - ln(f(x) - \dots))) + 3$
So, I've gotten to this point, but I'm not sure how to proceed- or whether or not I'd even need to proceed any further. Is a recursive answer like this one acceptable for such a question? If not, how can I try and express $f(x+3)$ in a non-recursive manner?
Thanks in advance.