Composition of functions, domain, range

[jaejoon89]

A = (0, infinity), B = C = D = R where R is all real numbers
f: A->B, g: B->C, h: C->D
f(x) = lnx, g(y) = 3y, h(z) = e^z
h o g o f ?

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For the following to be defined doesn't
1) range(f) ⊆ domain(g)
2) range(g o f) ⊆ domain(h)

So g o f should be defined since R ⊆ R and h o (g o f) should be defined since R ⊆ R.

But I don't understand how can you have the function h with the range of all real numbers when the exponential function only has a range of all positive real numbers?

So, what will the domain of the result be?

h(g(f(x)) = x^2 , all reals (?)

 

[Mark44]

Your formula h(g(f(x))) is incorrect. Work it out again without simplifying it.