Any linear function should satisfy the addition property:
$f(u + v) = f(u) + f(v)$
$au + av + b = au + av + 2b$
$b = 0$
Thus $f(u + v) = f(u) + f(v)$ only when $b = 0$ (the same is true for multiplication)
MilindNilekani
Well, building off of graphix has mentioned, neither of the primary rules for a linear function apply to f(x)=ax+b if b!=0. f(cx)=acx+b while cf(x)=acx+cb which is equal only for c=1 or b=0. However, this property should be fulfilled for all c belonging to R. So, we eliminate the other answer, therefore b=0 is the lone answer for this function to be linear function.
keenan
@MilindNilekani Yep, terrific. This is exactly the right argument.
It's a linear function only if $b = 0$
Any linear function should satisfy the addition property:
$f(u + v) = f(u) + f(v)$
$au + av + b = au + av + 2b$
$b = 0$
Thus $f(u + v) = f(u) + f(v)$ only when $b = 0$ (the same is true for multiplication)
Well, building off of graphix has mentioned, neither of the primary rules for a linear function apply to f(x)=ax+b if b!=0. f(cx)=acx+b while cf(x)=acx+cb which is equal only for c=1 or b=0. However, this property should be fulfilled for all c belonging to R. So, we eliminate the other answer, therefore b=0 is the lone answer for this function to be linear function.
@MilindNilekani Yep, terrific. This is exactly the right argument.