Symmetric Graphs: f(x)=3^x and g(x)=(1/3)^x Explained

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SUMMARY

The discussion clarifies the symmetry between the functions f(x)=3^x and g(x)=(1/3)^x, establishing that they mirror each other across the y-axis. By denoting g(x) as 3^{-x}, it is confirmed that for every point (x, y) on the graph of f, there exists a corresponding point (-x, y) on the graph of g. This mathematical relationship highlights the concept of symmetry in exponential functions, specifically in the context of their graphical representations.

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Fernando Revilla
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I quote a question from Yahoo! Answers

f(x)=3^x and g(x)=(1/3)^x I put that they mirror each other, that they are symmetrical. I am obviously missing something important between the two

I have given a link to the topic there so the OP can see my response.
 
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Denote $f(x)=3^x$ and $g(x)=(1/3)^x=1/3^x=3^{-x}$ and $\Gamma (f)$, $\Gamma (g)$ their respective graphs. Then, $$(x,y)\in\Gamma (f)\Leftrightarrow y=3^x \Leftrightarrow y=3^{-(-x)}\Leftrightarrow (-x,y)\in \Gamma (g)$$ This means that $f$ and $g$ are symmetrical with respect to the $y$-axis.
 

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