# Thread: Linear Function

1. ## Linear Function

I would appreciate help with the following problem please.
What values of c in the function f : R to R given by f(x) =mx + c, is f linear.

2. ## Re: Linear Function

Originally Posted by mlg
I would appreciate help with the following problem please.
What values of c in the function f : R to R given by f(x) =mx + c, is f linear.
QUESTION: Do you know the definition of linear function?
If so there is a place to start.
If not then look it up.

3. ## Re: Linear Function

Any function of the form f(x)= ax+ c is linear for any numbers a and c.

4. ## Re: Linear Function

Originally Posted by HallsofIvy
Any function of the form f(x)= ax+ c is linear for any numbers a and c.
Actually there are two uses for the term linear function.
Moreover, this question was posted in the number theory forum which suggests the second meaning to me.

5. ## Re: Linear Function

Thank you Plato & Hallsoflvy for the quick reply.
I will now look up the definition of linear function.

6. ## Re: Linear Function

The definition of linear function is:
A linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication:
f(x + y) = f(x) + f(y)
f(ax) = af(x)

How does this definition tie in with the value of c?

7. ## Re: Linear Function

Originally Posted by mlg
The definition of linear function is:
A linear function is a map f between two vector spaces that preserves vector addition and scalar multiplication:
f(x + y) = f(x) + f(y)
f(ax) = af(x)
How does this definition tie in with the value of c?
$f(x)=ax+c\\f(y)=ay+c\\f(x+y)=a(x+y)+c\\f(x)+f(y)= ~?$

8. ## Re: Linear Function

Thanks again Plato.

9. ## Re: Linear Function

To underscore this, I claim for a linear function $f: \Bbb R \to \Bbb R$ we must have:

$f(0) = 0$.

By LINEARITY:

$f(x + y) = f(x) + f(y)$. Taking $x = y = 0$, we get:

$f(0) = f(0+0) = f(0) + f(0)$ so that:

$0 = f(0) - f(0) = (f(0) + f(0)) - f(0) = f(0)) + (f(0) - f(0)) = f(0) + 0 = f(0)$.

Now if $f(x) = mx + c$, what is $f(0)$, and what does that tell you about what $c$ MUST be for $f(x) = mx + c$ to be linear?

10. ## Re: Linear Function

Thanks Deveno for your time and effort.

11. ## Re: Linear Function

Thanks Plato for your response.
I made the following attempt:
f(x) = ax + c
f(y) = ay + c
f(x + y) = a(x + y) + c
f(x) + f(y) = ax + c + ay + c = a(x + y) + 2c.
But a(x + y) + c is not equal to a(x + y) +2c.
i.e. c is not equal to 2c, unless c = 0.
(The question mentioned values, but c = 0 is only one value).
I would appreciate your comment.