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

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The functions f(x)=3^x and g(x)=(1/3)^x are indeed symmetrical with respect to the y-axis. This symmetry is demonstrated by the relationship where g(x) can be expressed as g(x)=3^{-x}, indicating that for every point (x,y) on the graph of f, there is a corresponding point (-x,y) on the graph of g. The discussion clarifies that both functions mirror each other, confirming their symmetrical nature. Understanding this relationship is crucial for analyzing their graphs effectively. The conclusion emphasizes the importance of recognizing the symmetry in these exponential functions.
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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