Problem 83 - College Algebra Essentials

Question

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 \(x=3\)

Step-by-Step Solution

Answer

The statement does not make sense because the horizontal asymptote is \(y=0\), not \(x=3\). Moreover, a horizontal asymptote would be a constant value of \(y\), not \(x\).

Step 1: Understand the Function and Asymptote

We begin by understanding the given function and asymptote. The function \(f(x)=3\cdot 2^{x}\) is an exponential function where the base (2) is positive. The horizontal asymptotes of exponential functions are lines that the curve approaches as \(x\) approaches plus or minus infinity. They occur at the limit of the function as \(x\) approaches infinity or negative infinity.

Step 2: Identifying the Asymptote

In the case of the function \(f(x)=3\cdot 2^{x}\), as \(x\) goes to negative infinity, the value of the function approaches zero. This happen because as \(x\) becomes increasingly negative, \(2^{x}\) approaches zero and thus, \(3\cdot 2^{x}\) also approaches zero. So the horizontal asymptote of our function is \(y=0\).

Step 3: Validating the Statement

The statement provided says the horizontal asymptote for \(f\) is \(x=3\). This is incorrect - asymptotes are lines and should be defined via the \(y\) variable, not \(x\). Secondly, it isn't \(y=3\), the horizontal asymptote for this function is \(y=0\).