Question

Find a function for which f(x+3) is equivalent to f(x) + f(3).

Answer 1

I've found a function that is NOT equivalent to f(x) + f(3). f(x) = (x-1) (x-3) (x+2) but I don't know how to find one that is equivalent.

Answer 2

Claim:
If f(x) = m*x, where m is any real number, then f(x+k) = f(x)+f(k)


Proof of that claim


f(x) = m*x
f(x+k) = m*(x+k)
f(x+k) = m*x+m*k

f(x) = m*x
f(k) = m*k

f(x)+f(k) = m*x + m*k

which means
f(x+k) = f(x)+f(k)
turns into
m*x+m*k = m*x + m*k
essentially telling us we have an identity

Answer 3

So essentially, if you have a linear equation going through the origin (0,0), then it will satisfy the property that f(x+3) = f(x) + f(3)

Answer 4

@jim_thompson5910 I see that now. Thank you!

Answer 5

no problem