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February 12th, 2019, 02:59 AM   #1
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If a function $f$ is inverse of another function $g$, then domain and codomain of $f$ are codomain and domain of $g$ respectively. My book has written that exponential function $a^x$ has codomain $R$, but since logarithmic function is its inverse then why logarithmic function's domain doesn't matches with codomain of $a^x$?

Last edited by skipjack; February 12th, 2019 at 01:23 PM.
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February 12th, 2019, 04:30 AM   #2
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There are exceptions to that rule. Consider, for example, $y=x^2$.
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February 12th, 2019, 12:42 PM   #3
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You are better off using range rather than codomain.
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February 12th, 2019, 03:34 PM   #4
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Quote:
Originally Posted by suyogya View Post
If a function $f$ is inverse of another function $g$, then domain and codomain of $f$ are codomain and domain of $g$ respectively. My book has written that exponential function $a^x$ has codomain $R$, but since logarithmic function is its inverse then why logarithmic function's domain doesn't matches with codomain of $a^x$?
This is wrong. The correct statement is the following:

If $f,g$ are inverses, then the domain of $f$ is the image of $g$ and the domain of $g$ is the image of $f$.

With the corrected version it is easily checked for your example. The (maximum) domain for $\log(x)$ is $(0,\infty)$ and as expected, the image of $e^x$ on the domain $\mathbb{R}$ is exactly $(0,\infty)$.
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