Istoria Bisericii Orthodoxe Romane Mircea Pacurariu.pdf 2021
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Need help with a few things about this proof.
The following is a question that I just got in a test I was taking on Tuesday. I did not understand any of the homework, so I am writing this for my homework as I do not understand how to approach this problem.
Question: Let $f: R to R$ be defined as $f(x)=e^x$.
Show that for any $a,b in R$, the number $f(a)+f(b)$ can have at most one value.
My work:
I am given the problem statement and asked to use the hint that $f(x)$ is the exponential function and use that it only has one value which is $1$ which I actually did. I was given the tip of $implies$ which I really did not understand and have no idea what that even means. The only thing I can think of is that since $f(x)$ is the exponential function, $f(a)+f(b)$ should be equal to $f(a+b)$ which I am not sure is true. I am assuming that the hint is actually $implies$.
In order to prove the question I am not sure how to approach it. It seems pretty vague. I am unsure whether to use a truth table or the fact that we can find a $c$ where $f(x)=f(c)$.
I am able to find that $f(x)$ is a strictly increasing function and also a strictly decreasing function. I am also able to find that $f(0)$=0 and $f(1)$=1. I am thinking that I need to use the expression $f(a)+f(b)$ but I am not sure how to continue.
I would really appreciate if someone could offer a hint or answer for this.
A:
You may have $a + b gt 1$ and $a + b lt 1$. You may have $a + b gt 1$ and $a + b gt 1$.
You may want to recall the exponential function’s properties.
[Hint : Show
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