- The domain of the exponential function \(h(x) = 2^{x+4}\) is the set of all real numbers, as any real value can be used as an exponent.
- The range consists of all values greater than 0, as a positive base raised to any power is always positive.
- The horizontal asymptote is the line \(y = 0\), which the function approaches as \(x \to -\infty\).
- Final answer: \(\text{domain: } \{x \mid x \text{ is a real number}\}; \text{ range: } \{y \mid y > 0\}; \text{ asymptote: } y = 0\).
Explanation
- Problem Analysis
The given function is \(h(x) = 2^{x+4}\). This is a standard exponential function of the form \(f(x) = b^{x+c} + d\), where \(b=2\), \(c=4\), and \(d=0\). To understand its behavior, we examine the domain, range, and horizontal asymptote.
- Determining the Domain
The domain of an exponential function \(b^{x+c}\) is the set of all possible input values \(x\) for which the function is defined. Since the exponent \((x+4)\) can be any real number, the function \(h(x) = 2^{x+4}\) is defined for all real numbers. Thus, the domain is \(\{x \mid x \in \mathbb{R}\}\), or the set of all real numbers.
- Determining the Range
For the range, consider the output of \(h(x) = 2^{x+4}\). Since the base \(2\) is positive, any power of \(2\) will always be greater than \(0\). As \(x\) approaches positive infinity, \(2^{x+4}\) grows without bound. As \(x\) approaches negative infinity, \(2^{x+4}\) becomes arbitrarily close to \(0\) but never reaches it. Therefore, the function outputs all values greater than \(0\). The range is \(\{y \mid y > 0\}\).
- Determining the Horizontal Asymptote
The horizontal asymptote represents the value the function approaches as \(x\) moves toward positive or negative infinity. As \(x \to -\infty\), the exponent \(x+4\) approaches \(-\infty\), meaning \(2^{x+4}\) approaches \(0\). Thus, the graph of \(h(x)\) has a horizontal asymptote at the line \(y = 0\).
- Conclusion and Final Answer
Based on our analysis, the domain is the set of all real numbers, the range is \(\{y \mid y > 0\}\), and the horizontal asymptote is \(y = 0\). Comparing this to the given choices, the correct description is: domain: \(\{x \mid x \text{ is a real number}\}\); range: \(\{y \mid y > 0\}\); asymptote: \(y = 0\).
Examples
Exponential functions like \(h(x) = 2^{x+4}\) are fundamental in modeling natural phenomena. For instance, if a colony of bacteria doubles every hour, and we shift the time scale by 4 hours, the population at time \(x\) can be represented by \(2^{x+4}\). Understanding the domain and range helps scientists predict that the population can be measured at any point in time (domain is all real numbers) and that the population will always be positive (range is \(y > 0\)). The horizontal asymptote \(y=0\) suggests that as we look further back in time, the population approaches zero, representing the start of the growth process.