Question: Draw the graph of \( f(x) = \left(\frac{1}{3}\right)^x \). [Graph grid with x-axis from -10…

Draw the graph of \( f(x) = \left(\frac{1}{3}\right)^x \).

[Graph grid with x-axis from -10 to 10 and y-axis from -10 to 10]

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Solution

The problem is to draw the graph of the function \( f(x) = \left( \frac{1}{3} \right)^x \). This is an exponential decay function with a base less than 1. Here’s how you can approach it: Identify key points to plot: 1. When \( x = 0 \), \( f(x) = 1 \). 2. When \( x = 1 \), \( f(x) = \frac{1}{3} \). 3. When \( x = 2 \), \( f(x) = \left( \frac{1}{3} \right)^2 = \frac{1}{9} \). 4. When \( x = -1 \), \( f(x) = 3 \) (because \( \left( \frac{1}{3} \right)^{-1} = 3 \)). 5. When \( x = -2 \), \( f(x) = 9 \). Plot these points on the graph. Draw a smooth curve that passes through these points, approaching the x-axis as \( x \) increases, showing that the function approaches zero but never actually reaches it. The graph will have a horizontal asymptote at \( y = 0 \). This will give you the complete graph of \( f(x) = \left( \frac{1}{3} \right)^x \).