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It should be clear that this function is not linear; take, say,Originally Posted by benjamin872
a linear function has to satisfy:
additivity
and homogeneity
but how do i show this for some simple function like f(x)=x+3??
i know it must be contained in the domain. but i want to be 100% with my workings im more confused with the whole homogeneous bit.then
and also, if we take, say,then
 = \alpha x + 3 = 2 * 1 + 3 = 5 \neq \alpha f(x) = \alpha(x+3) = 2(1+3) = 8)
You don't- as defunkt says, that function is NOT "linear" in this sense. The only linear functions on the real numbers are those of the form f(x)= ax for some a.a linear function has to satisfy:
additivity
and homogeneity
but how do i show this for some simple function like f(x)=x+3??
i know it must be contained in the domain. but i want to be 100% with my workings im more confused with the whole homogeneous bit.
ok but then il trysay 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.ok but then il try
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
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.
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