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Chapter 3: Problem 17

Use a graphing utility to construct a table of values for the function. Then sketch the graph of the function. $$f(x)=\left(\frac{1}{2}\right)^{x}$$

Short Answer

Expert verified
The graph of the function \(f(x) = \left(\frac{1}{2}\right)^{x}\) decreases as x increases, showing an asymptotic behavior towards the x-axis. When x decreases, the function \(f(x)\) increases.

Step by step solution

01

Construct the table of values

Utilize a graphing calculator or online tool to generate a table of values for the function \(f(x) = \left(\frac{1}{2}\right)^{x}\). Vary x from -3 to 3 to have an effective scope.
02

Sketching the Graph

Based on the table of values, begin by placing each point on a graph. Start from the leftmost point and slowly work towards the right, marking each value of x against the corresponding value of \(f(x)\). Join these points with a smooth curve.
03

Recognizing the Behavior

As visible from the graph, as x increases, \(f(x)\) gets closer and closer to zero but never actually reaches it. And as x decreases, \(f(x)\) gets larger. This shows that the function has a horizontal asymptote at y=0.

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