.
It should be clear that this function is not linear; take, say, $\displaystyle x=1, y=2$ then $\displaystyle f(x+y) = x+y+3 = 1+2+3 = 6 \neq f(x) + f(y) = x+3+y+3 = 1+3+2+3 = 9$
and also, if we take, say, $\displaystyle \alpha = 2$ then $\displaystyle f(\alpha x) = \alpha x + 3 = 2 * 1 + 3 = 5 \neq \alpha f(x) = \alpha(x+3) = 2(1+3) = 8$
ok but then il try $\displaystyle f(x)=2x+4 $ say for x=1 and y=2
i find f(x+y) not equal to f(x)+f(y) which makes 10 not equal 14. so this is not linear also? can someone give me an example of a linear function that works.
......... wait so the these properties cannot be satisfied if there is a constant, but i thought a linear function could be plotted y=mx+b im confused
Those are two different meanings of the word "linear". In algebra, a function of the form f(x)= mx+ b is called "linear" because its graph is a straight line.
But Linear Algebra uses a more stringent definition for "linear transformation"- we must have f(x+ y)= f(x)+ f(y) and f(ax)= af(x). As I said above, the only "linear" functions in that sense, from R to R are of the form f(x)= ax for some number a.