I'm taking a trip back to the beginnings of algebra, when we were told that a linear function is of the form

y=mx+b. But this can't be true, because I came across a definition for linear functions, saying that for a function to be considered linear

f(a+c) = f(a)+f(c)
m(a+c)+b = ma+b+mc+b
ma+mc+b certainly is not equal to ma+mc+2b

unless your y-intercept was 0.

So shouldn't the definition of a linear function be a constant function that passes through the origin? If so, why has my instructors lied to me until now!?!? WHY!?!?!?! :p

Well, I never saw that definition until now... I know that a linear function is always in the form y = mx + c (I'm used to have the letter c as constant)

I see that the definition is correct only if the y-intercept is zero (like you said), which is not the case for the majority of linear functions.

Also, I've learned that many teachers lie to us in small classes, whether in Maths, chemistry, or physics, or biology.

For example, the usual x^2 = -4 has no solutions. But no! it has solutions, but in complex numbers!

Well, I never saw that definition until now... I know that a linear function is always in the form y = mx + c (I'm used to have the letter c as constant)

I see that the definition is correct only if the y-intercept is zero (like you said), which is not the case for the majority of linear functions.

Also, I've learned that many teachers lie to us in small classes, whether in Maths, chemistry, or physics, or biology.

For example, the usual x^2 = -4 has no solutions. But no! it has solutions, but in complex numbers!

Well i mean there is the little lies, since technically complex numbers lie in a separate number plane all together, but I can't see the point of how practicality in the whole linear vs constant function definitions are warped into one. We clearly state that there is a difference between a mclaurin series and a taylor series, but not a constant/linear function? seems a bit contradictory to me.

As far as I'm aware a constant function just means that it's a horizontal straight line, that is, y always equals the same number. I see it as being a "sub-set" of linear functions, which are just straight lines.

Perhaps "constant" function was the wrong term, but rather a function whose slope is constant.

I'm taking a trip back to the beginnings of algebra, when we were told that a linear function is of the form

y=mx+b. But this can't be true, because I came across a definition for linear functions, saying that for a function to be considered linear

f(a+c) = f(a)+f(c)
m(a+c)+b = ma+b+mc+b
ma+mc+b certainly is not equal to ma+mc+2b

unless your y-intercept was 0.

So shouldn't the definition of a linear function be a constant function that passes through the origin? If so, why has my instructors lied to me until now!?!? WHY!?!?!?! :p

Hello Nhatkeim:

I think most of use the definition of a linear function as being a first order polynomial, in other words f(x) = mx + b. This is how linear functions are defind in Linear function - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Linear_function)

However, I found a definition on the web that broadens this - from linear function - Wiktionary (http://en.wiktionary.org/wiki/linear_function) :

1. Any function whose graph is a straight line
2. Any function of the sum of two variables whose value is the same as the sum of the values of the same function of the two variables singly

That first definition is equivalent to what Wikipedia says, and the second is equivalent to the alternate definition you cited. I think the right way to read this is that if a function meets either of these critria then it can be considered as linear, but it is not required that the function meet both definitions.

I just find it strange that the higher level math I've been taking recently has pretty much on some level redefining the definitions of things I had already learn that have been carved neatly into my brain. It's interesting, but at the same time, quiet the annoyance.