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:

Analyze the domain of the function

Using the Domain and Range and Exponential Functions knowledge points

$$
LATEXBLOCK0
$$

Determine the range of the function

Using the Exponential Range and Function Transformations knowledge points

$$
LATEXBLOCK1
$$

Find the horizontal asymptote

Using the Exponential Functions knowledge point

$$
LATEXBLOCK2
$$

Answer:

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