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

In summary, the graphs of the functions $f(x)=3^x$ and $g(x)=(1/3)^x$ are symmetrical with respect to the $y$-axis.
  • #1
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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  • #2
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.
 

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

What is a symmetric graph?

A symmetric graph is a graph in which the left and right sides are mirror images of each other. This means that if a point (x,y) is on the graph, then the point (-x,y) is also on the graph.

How are f(x)=3^x and g(x)=(1/3)^x symmetric?

These two functions are symmetric because the exponent in each function can be changed to its opposite sign without changing the overall shape of the graph. This means that f(x) and g(x) will have the same y-values for opposite x-values, making them mirror images of each other.

How can I graph f(x)=3^x and g(x)=(1/3)^x?

To graph these functions, you can plot several points and then connect them with a smooth curve. You can also use a graphing calculator or online graphing tool to quickly and accurately graph the functions.

What is the domain and range of f(x)=3^x and g(x)=(1/3)^x?

The domain of both functions is all real numbers, as there are no restrictions on the x-values. The range of f(x)=3^x is all positive real numbers, while the range of g(x)=(1/3)^x is all positive real numbers less than 1.

How can I determine if a graph is symmetric without graphing it?

To determine if a graph is symmetric, you can evaluate the function for opposite x-values. If the resulting y-values are the same, then the graph is symmetric. You can also check if the function is even or odd, as even functions are symmetric about the y-axis and odd functions are symmetric about the origin.

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