Question

what are the domain, range, and asymptote of (h(x) = 2^{x + 4})?

domain: ({x mid x > 0}); range: ({y mid y \text{ is a real number}}); asymptote: (y = 0)
domain: ({x mid x > -4}); range: ({y mid y \text{ is a real number}}); asymptote: (y = -4)
domain: ({x mid x \text{ is a real number}}); range: ({y mid y > 0}); asymptote: (y = 0)
domain: ({x mid x \text{ is a real number}}); range: ({y mid y > 0}); asymptote: (y = -4)

Explanation:

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<reasoning>

Analyze the domain of the function

Using the Domain and Range and Exponential Functions knowledge points
\[

$$\begin{aligned} &\text{For } h(x) = 2^{x + 4}\\ &\text{Domain: } \{x \mid x \text{ is a real number}\} \end{aligned}$$

\]

Determine the range of the function

Using the Exponential Range and Function Transformations knowledge points
\[

$$\begin{aligned} &2^{x+4} > 0 \quad \forall x \in \mathbb{R}\\ &\text{Range: } \{y \mid y > 0\} \end{aligned}$$

\]

Find the horizontal asymptote

Using the Exponential Functions knowledge point
\[

$$\begin{aligned} &\lim_{x \to -\infty} 2^{x+4} = 0\\ &\text{Asymptote: } y = 0 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>domain: \(\{x \mid x > 0\}\); range: \(\{y \mid y \text{ is a real number}\}\); asymptote: \(y = 0\)</mcq-option>
<mcq-option>domain: \(\{x \mid x > -4\}\); range: \(\{y \mid y \text{ is a real number}\}\); asymptote: \(y = -4\)</mcq-option>
<mcq-correct>domain: \(\{x \mid x \text{ is a real number}\}\); range: \(\{y \mid y > 0\}\); asymptote: \(y = 0\)</mcq-correct>
<mcq-option>domain: \(\{x \mid x \text{ is a real number}\}\); range: \(\{y \mid y > 0\}\); asymptote: \(y = -4\)</mcq-option>
</answer>

<plot>
{
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"type": "functiongraph",
"params": [
{
"js": "Math.pow(2, x + 4)",
"latex": "2^{x+4}"
},
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2
],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 3,
"name": "h(x) = 2^{x+4}",
"withLabel": true
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Answer:

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<reasoning>

Analyze the domain of the function

Using the Domain and Range and Exponential Functions knowledge points
\[

$$\begin{aligned} &\text{For } h(x) = 2^{x + 4}\\ &\text{Domain: } \{x \mid x \text{ is a real number}\} \end{aligned}$$

\]

Determine the range of the function

Using the Exponential Range and Function Transformations knowledge points
\[

$$\begin{aligned} &2^{x+4} > 0 \quad \forall x \in \mathbb{R}\\ &\text{Range: } \{y \mid y > 0\} \end{aligned}$$

\]

Find the horizontal asymptote

Using the Exponential Functions knowledge point
\[

$$\begin{aligned} &\lim_{x \to -\infty} 2^{x+4} = 0\\ &\text{Asymptote: } y = 0 \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>domain: \(\{x \mid x > 0\}\); range: \(\{y \mid y \text{ is a real number}\}\); asymptote: \(y = 0\)</mcq-option>
<mcq-option>domain: \(\{x \mid x > -4\}\); range: \(\{y \mid y \text{ is a real number}\}\); asymptote: \(y = -4\)</mcq-option>
<mcq-correct>domain: \(\{x \mid x \text{ is a real number}\}\); range: \(\{y \mid y > 0\}\); asymptote: \(y = 0\)</mcq-correct>
<mcq-option>domain: \(\{x \mid x \text{ is a real number}\}\); range: \(\{y \mid y > 0\}\); asymptote: \(y = -4\)</mcq-option>
</answer>

<plot>
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"params": [
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