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February 12th, 2019, 02:59 AM  #1 
Newbie Joined: Feb 2019 From: home Posts: 5 Thanks: 0  mathematics help
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. 
February 12th, 2019, 04:30 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,929 Thanks: 1124 Math Focus: Elementary mathematics and beyond 
There are exceptions to that rule. Consider, for example, $y=x^2$.

February 12th, 2019, 12:42 PM  #3 
Global Moderator Joined: May 2007 Posts: 6,728 Thanks: 689 
You are better off using range rather than codomain.

February 12th, 2019, 03:34 PM  #4  
Senior Member Joined: Sep 2016 From: USA Posts: 598 Thanks: 366 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
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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