Question

which function has a horizontal asymptote of \\(y = 3\\)?

\\(f(x) = 3(2^x)\\)
\\(f(x) = 2(4)^{x-3}\\)
\\(f(x) = 2(3^x)\\)
\\(f(x) = 2(4^x) + 3\\)

Explanation:

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<reasoning>

Analyze the standard horizontal asymptote of exponential functions

Using the Exponential Transformations knowledge point
\[

$$\begin{aligned} &g(x) = a \cdot b^x\\ &\lim_{x \to -\infty} g(x) = 0 \quad (\text{for } b > 1)\\ &\text{Horizontal Asymptote: } y = 0 \end{aligned}$$

\]

Apply vertical translation to find the shifted asymptote

Using the Function Translation and Vertical Translation knowledge points
\[

$$\begin{aligned} &f(x) = a \cdot b^x + k\\ &\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (a \cdot b^x + k) = 0 + k = k\\ &\text{Horizontal Asymptote: } y = k \end{aligned}$$

\]

Evaluate the given options for \(k = 3\)

Using the Vertical Translation knowledge point
\[

$$\begin{aligned} &f(x) = 3(2^x) \implies k = 0 \implies y = 0\\ &f(x) = 2(4)^{x-3} \implies k = 0 \implies y = 0\\ &f(x) = 2(3^x) \implies k = 0 \implies y = 0\\ &f(x) = 2(4^x) + 3 \implies k = 3 \implies y = 3 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(f(x) = 3(2^x)\)</mcq-option>
<mcq-option>(B) \(f(x) = 2(4)^{x-3}\)</mcq-option>
<mcq-option>(C) \(f(x) = 2(3^x)\)</mcq-option>
<mcq-correct>(D) \(f(x) = 2(4^x) + 3\)</mcq-correct>
</answer>

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Answer:

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<reasoning>

Analyze the standard horizontal asymptote of exponential functions

Using the Exponential Transformations knowledge point
\[

$$\begin{aligned} &g(x) = a \cdot b^x\\ &\lim_{x \to -\infty} g(x) = 0 \quad (\text{for } b > 1)\\ &\text{Horizontal Asymptote: } y = 0 \end{aligned}$$

\]

Apply vertical translation to find the shifted asymptote

Using the Function Translation and Vertical Translation knowledge points
\[

$$\begin{aligned} &f(x) = a \cdot b^x + k\\ &\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (a \cdot b^x + k) = 0 + k = k\\ &\text{Horizontal Asymptote: } y = k \end{aligned}$$

\]

Evaluate the given options for \(k = 3\)

Using the Vertical Translation knowledge point
\[

$$\begin{aligned} &f(x) = 3(2^x) \implies k = 0 \implies y = 0\\ &f(x) = 2(4)^{x-3} \implies k = 0 \implies y = 0\\ &f(x) = 2(3^x) \implies k = 0 \implies y = 0\\ &f(x) = 2(4^x) + 3 \implies k = 3 \implies y = 3 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(f(x) = 3(2^x)\)</mcq-option>
<mcq-option>(B) \(f(x) = 2(4)^{x-3}\)</mcq-option>
<mcq-option>(C) \(f(x) = 2(3^x)\)</mcq-option>
<mcq-correct>(D) \(f(x) = 2(4^x) + 3\)</mcq-correct>
</answer>

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