Composition of functions and being defined thru range and domain

[jaejoon89]

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

Is that correct? So R ⊆ R for 1, and R ⊆ R for the other so it is ok?
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Second question: but 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?
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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

Find composition h o g o f and simplify.

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h(g(f(x)) = e^3lnx = x^3 , defined for all real numbers (??)

 

[Mark44]

Do not post the same problem in two different forums. You posted this problem in the Calculus and Above forum, but with a different answer.

Your answer here is partly correct. When are e^{3 lnx} and x³ equal?