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?

 

[Architeuthis Dux]

I can confirm that your understanding of composition of functions and their domains and ranges is correct. In order for a function to be defined, the range of the previous function must be a subset of the domain of the next function. In the first question, both ranges are subsets of the same domain, so the composition is valid.

In the second question, it is important to note that the range of a function is not limited to the values it can output. It is simply the set of all possible output values. In the case of the exponential function, although it is commonly used to represent only positive real numbers, it is defined for all real numbers. Therefore, the composition h o g o f is valid for all real numbers.

To simplify the composition, we can use the property of logarithms and exponents to rewrite e^3lnx as (e^lnx)^3. Since e^lnx is equal to x, the composition simplifies to x^3.