Answered

Which function has a horizontal asymptote of $y=3$?


A$f(x)=3\left(2^x\right)$
B$f(x)=2(4)^{x-3}$
C$f(x)=2\left(3^x\right)$
D$f(x)=2\left(4^x\right)+3$

Verified Answer :

  • Identify that a horizontal asymptote \(y=k\) for an exponential function \(f(x) = a \cdot b^{g(x)} + k\) occurs as the exponential term approaches zero.
  • Analyze the provided functions and note that the first three, \(f(x)=3(2^x)\), \(f(x)=2(4)^{x-3}\), and \(f(x)=2(3^x)\), all have a constant term \(k=0\), resulting in a horizontal asymptote of \(y=0\).
  • Recognize that the function \(f(x)=2(4^x)+3\) has a constant term \(k=3\), meaning its horizontal asymptote is \(y=3\).
  • Conclude that the correct function is \(\boxed{f(x)=2(4^x)+3}\).

Explanation

  1. Understanding Horizontal Asymptotes for Exponential Functions

A horizontal asymptote of a function \(f(x)\) is a horizontal line \(y = L\) that the graph of the function approaches as \(x\) approaches positive or negative infinity. For a general exponential function of the form \(f(x) = a \cdot b^{g(x)} + k\), where \(a, b,\) and \(k\) are constants and \(g(x)\) is a linear function (like \(x\) or \(x-c\)), the behavior is determined by the term \(a \cdot b^{g(x)}\). As \(x\) moves toward \(\infty\) or \(-\infty\), the term \(a \cdot b^{g(x)}\) will approach \(0\), provided the base \(b > 0\) and \(b eq 1\). Consequently, the function value \(f(x)\) will approach the constant \(k\). Therefore, the horizontal asymptote is \(y = k\).

  1. Evaluating Each Function

We examine each of the four provided functions to determine their horizontal asymptotes:

  1. \(f(x) = 3(2^x)\): This is of the form \(a \cdot b^x + k\) with \(k=0\). As \(x \to -\infty\), \(2^x \to 0\), so \(f(x) \to 0\). The horizontal asymptote is \(y=0\).
  2. \(f(x) = 2(4)^{x-3}\): This can be written as \(f(x) = 2 \cdot 4^x \cdot 4^{-3} = \frac{2}{64} \cdot 4^x + 0\). This is of the form \(a \cdot b^x + k\) with \(k=0\). As \(x \to -\infty\), \(4^x \to 0\), so \(f(x) \to 0\). The horizontal asymptote is \(y=0\).
  3. \(f(x) = 2(3^x)\): This is of the form \(a \cdot b^x + k\) with \(k=0\). As \(x \to -\infty\), \(3^x \to 0\), so \(f(x) \to 0\). The horizontal asymptote is \(y=0\).
  4. \(f(x) = 2(4^x) + 3\): This is of the form \(a \cdot b^x + k\) with \(k=3\). As \(x \to -\infty\), \(4^x \to 0\), so \(f(x) \to 2(0) + 3 = 3\). The horizontal asymptote is \(y=3\).
  1. Conclusion and Final Selection

Comparing our analysis with the target asymptote of \(y=3\), we see that only the function \(f(x) = 2(4^x) + 3\) satisfies this condition. The other three functions have a horizontal asymptote of \(y=0\). Therefore, the function that has a horizontal asymptote of \(y=3\) is \(f(x) = 2(4^x) + 3\).

Examples

In the world of pharmacology, the concentration of a medication in the bloodstream over time is often modeled using exponential functions. If a medication is being administered continuously and the body metabolizes it, the concentration might follow a curve that approaches a steady-state level. If the concentration \(C(t)\) of a drug is given by \(C(t) = C_0 \cdot e^{-kt} + S\), the value \(S\) represents the horizontal asymptote, which is the steady-state concentration the drug levels out to after a long period. Understanding horizontal asymptotes helps doctors predict these steady states to ensure patient safety and medication efficacy.

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