Is it true that for all functions f and g, if f ºg=2x then the domain of f ºg is all real numbers?

Answer 1

Hi Naeema,

f^og=2x means that f(g(x))=2x

So, maybe f(x)=4x and g(x)=0.5x for example. This would give f(g(x))=4(0.5x)=2x.

The domain is all real numbers unless you have a possibility to divide by zero, or to take square root of a negative number, etc.

So, is it possible for f(x) or g(x) to have one of these issues like dividing by zero?

I think so.

Consider this:

f(x)=2/x

g(x)=1/x

In this case, f(g(x))=2/(1/x)=2x

While the domain is all real numbers for y=2x, it is actually NOT all real numbers for f(g(x))=2x necessarily...

Because it is a composite function, it is possible for the g(x) to have a domain that excludes zero, for example, and then the domain of the composite function will also exclude zero.

So no, it is not true to say for ALL functions f and g, if f^og=2x then the domain of f^og is all real numbers.

Hope this helps! :)