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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: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond There are exceptions to that rule. Consider, for example, $y=x^2$. Thanks from Denis and topsquark February 12th, 2019, 12:42 PM #3 Global Moderator   Joined: May 2007 Posts: 6,823 Thanks: 723 You are better off using range rather than codomain. Thanks from topsquark February 12th, 2019, 03:34 PM   #4
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Quote:
 Originally Posted by suyogya 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 $(0,\infty)$. Tags codomain, domain, function, mathematics Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Navina Calculus 3 September 27th, 2014 05:48 AM sahil885 Math Books 3 April 11th, 2014 05:53 PM r-soy Calculus 3 February 9th, 2013 02:39 PM r-soy Calculus 2 February 8th, 2013 01:31 AM

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