Problem 6

A function

f:\mathbb Q\to\mathbb Q

is called aquaesulian if the following property holds: for every

x,y\in\mathbb Q

f(x+f(y))=f(x)+y

f(f(x)+y)=x+f(y)

. Show that there exists an integer

c

such that for any aquaesulian function

f

there are at most

c

different rational numbers of the form

f(r)+f(-r)

for some rational number

r

, and find the smallest possible value of

c

Step 3 of 5: f is injective

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In plain words

If two inputs gave the same output, the relation applied in whichever direction holds between them would immediately force the inputs to be equal, since the extra term added is exactly the difference between the two inputs.

f(a)=f(b)\implies a=b

Detailed analysis

Suppose $f(a)=f(b)$ ; without loss of generality $a\to b$ , so $f(a)+a=f(a+f(a))=f(a+f(b))=f(a)+b$ , giving $a=b$ . Hence $f$ is injective.