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Chapter 3: Problem 83

Determine whether each statement makes sense or does not make sense, and explain your reasoning. My graph of \(f(x)=3 \cdot 2^{x}\) shows that the horizontal asymptote for \(f\) is \(x=3\)

Short Answer

Expert verified
The statement does not make sense because it incorrectly states the horizontal asymptote as \(x=3\). The horizontal asymptote of the function \(f(x)=3 \cdot 2^{x}\) is \(y=0\).

Step by step solution

01

Identify the function

The function given is \(f(x)=3 \cdot 2^{x}\), which is an exponential function with base 2. Mentioning the base is important as it gives insights about the function behavior. Since the base (2) is greater than 1, we know the function is increasing and its graph will be rising as we move from left to right.
02

Determine the horizontal asymptote

The horizontal asymptote of a function is the value that the function approaches as it goes to positive or negative infinity. For an exponential function with a positive base like ours, the function will approach 0 as \(x\) goes to negative infinity. Thus, the horizontal asymptote is \(y=0\)
03

Analyze the statement

The statement says that the horizontal asymptote of the function is \(x=3\). This doesn't make sense, as a horizontal asymptote should be a value that \(y\) approaches, not \(x\). The correct way to represent a horizontal asymptote is \(y=k\), in our case, it is \(y=0\)

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