- A horizontal asymptote exists where \(\lim_{x \to \infty} f(x) = L\) or \(\lim_{x \to -\infty} f(x) = L\).
- Analyze the behavior of each function as \(x\) approaches \(\pm\infty\).
- Observe that for \(f(x) = 2(4^x) + 3\), the term \(4^x\) approaches 0 as \(x \to -\infty\).
- Conclude that \(\lim_{x \to -\infty} (2(4^x) + 3) = 3\), confirming the horizontal asymptote is \(\boxed{f(x)=2(4^x)+3}\).
Explanation
- Understanding Horizontal Asymptotes
A horizontal asymptote of a function \(f(x)\) is a line \(y=L\) that the graph approaches as \(x\) approaches positive or negative infinity. Formally, this occurs if \(\lim_{x \to \infty} f(x) = L\) or \(\lim_{x \to -\infty} f(x) = L\). We examine each of the four given functions to determine which one approaches 3 as \(x\) tends to infinity or negative infinity. The functions are:
- \(f(x) = 2(3^x)\)
- \(f(x) = 3(2^x)\)
- \(f(x) = 2(4)^{x-3}\)
- \(f(x) = 2(4^x) + 3\)
- Evaluating Limits of the Functions
We evaluate the limits for each function:
- For \(f(x) = 2(3^x)\): As \(x \to \infty\), \(3^x \to \infty\), so \(f(x) \to \infty\). As \(x \to -\infty\), \(3^x \to 0\), so \(f(x) \to 2(0) = 0\). This function has a horizontal asymptote of \(y=0\).
- For \(f(x) = 3(2^x)\): As \(x \to \infty\), \(2^x \to \infty\), so \(f(x) \to \infty\). As \(x \to -\infty\), \(2^x \to 0\), so \(f(x) \to 3(0) = 0\). This function has a horizontal asymptote of \(y=0\).
- For \(f(x) = 2(4)^{x-3}\): As \(x \to \infty\), \(4^{x-3} \to \infty\), so \(f(x) \to \infty\). As \(x \to -\infty\), \(4^{x-3} \to 0\), so \(f(x) \to 2(0) = 0\). This function has a horizontal asymptote of \(y=0\).
- For \(f(x) = 2(4^x) + 3\): As \(x \to \infty\), \(4^x \to \infty\), so \(f(x) \to \infty\). As \(x \to -\infty\), \(4^x \to 0\), so \(f(x) \to 2(0) + 3 = 3\). This function has a horizontal asymptote of \(y=3\).
- Conclusion and Final Answer
Comparing the results, only the fourth function, \(f(x) = 2(4^x) + 3\), approaches the value 3 as \(x\) approaches negative infinity. Therefore, it is the only function among the choices that possesses a horizontal asymptote at \(y=3\). The final answer is \(\boxed{f(x)=2(4^x)+3}\).
Examples
Horizontal asymptotes are frequently used in population modeling to represent carrying capacity—the maximum population size an environment can sustain. For example, if a population grows according to \(P(t) = 1000(1-e^{-kt}) + 500\), the term \(1000(1-e^{-kt})\) approaches 1000 as \(t \to \infty\), meaning the population stabilizes near \(1000+500=1500\). Identifying these asymptotes helps biologists and urban planners predict long-term trends in growth and sustainability.