if g(2x)=2g(x) prove that g(x) is a linear function, if it is true.?

Answer 1

If g(2x)=2g(x), then it a linear function.
Divide both sides of the equation by g(x), then g(2x)/g(x)=g(x),
2g(x)/g(x)=2, the following equation is then obtained.
g(x)=2. since g(x) is equal to 2 (a constant) then the function is linear becase any point you put in to g(x) always lies on that same line, it can't be any other function such as quadratic, polynomial or rational because those functions would also represent negetive values and have different critical points. Also it is linear because a constant calue doesn't really exist in functions with curves.

Hope this helps.

Answer 2

It's not true, as long as g doesn't have to be continuous.

Define g(x) like this:
g(x) = x if x is rational
g(x) = 0 if x is irrational

This satisfies the equality, since it satisfies it separately for all the rational and all the irrational numbers. Note that 2x is rational iff x is, so the equality will never lead us to compare g() for a rational and an irrational number.

But g() is not linear.

Answer 3

given: the derivative of a linear function is constant

d/dx g(2x) = 2g'(2x)
d/dx 2g(x) = 2g'(x)

since g(2x) = 2g(x) then d/dx g(2x) = d/dx 2g(x)

so 2g'(2x) = 2g'(x)
g'(2x) = g'(x)

this shows that the derivative of any argument to the function is the same as any other point. therefore the function is linear

Answer 4

Not only is the function linear, its graph is through the origin. if g(x) has a point (x,y) on the graph it also must have the point (2x,2y) on the graph. This is a dilation. The only graphs that are unchanged by a dilation are lines through the origin.