- The domain of the exponential function \(h(x) = (1.4)^x + 5\) is all real numbers, as there are no restrictions on the exponent \(x\).
- The range is \(\{y \mid y > 5\}\) because the base exponential part \((1.4)^x\) is always positive, and adding 5 shifts the entire range up.
- The horizontal asymptote is \(y = 5\), representing the value that the function approaches but never reaches as \(x\) decreases toward negative infinity.
- The correct option is: \(\boxed{\text{domain: } \{x \mid x \text{ is a real number}\}; \text{range: } \{y \mid y > 5\}; \text{asymptote: } y = 5}\).
Explanation
- Determine the Domain
The function is given by \(h(x) = (1.4)^x + 5\). To understand its properties, we analyze the basic exponential function \(f(x) = (1.4)^x\). An exponential function of the form \(b^x\) where \(b > 0\) and \(b
eq 1\) is defined for all real numbers. Since the addition of a constant (in this case, 5) does not impose any restrictions on the input values, the domain of \(h(x)\) is all real numbers.
- Determine the Range
The basic exponential function \(f(x) = (1.4)^x\) has a range of \((0, \infty)\) because \(b^x > 0\) for any real \(x\) and any base \(b > 0\). The function \(h(x) = (1.4)^x + 5\) is a vertical shift of \(f(x)\) upwards by 5 units. Thus, the range is shifted by 5, resulting in \((0+5, \infty+5)\), which simplifies to \((5, \infty)\). In set-builder notation, this is \(\{y \mid y > 5\}\).
- Identify the Horizontal Asymptote
The horizontal asymptote of the parent function \(f(x) = (1.4)^x\) occurs as \(x\) approaches \(-\infty\), where \(f(x)\) approaches 0. Therefore, the line \(y = 0\) is the horizontal asymptote of \(f(x)\). Applying the same vertical shift of 5 units to the horizontal asymptote, we find that the new horizontal asymptote for \(h(x) = (1.4)^x + 5\) is \(y = 0 + 5\), which is \(y = 5\).
- Conclusion
Comparing our findings to the given options:
- Domain: \(\{x \mid x \text{ is a real number}\}\)
- Range: \(\{y \mid y > 5\}\)
- Asymptote: \(y = 5\)
These properties match the first provided option. Thus, the correct choice is:
domain: \(\{x \mid x \text{ is a real number}\}\); range: \(\{y \mid y > 5\}\); asymptote: \(y = 5\).
Examples
In real-world applications, exponential functions are frequently used to model growth processes. For example, if you place money in a savings account with interest that compounds over time, the value of your account can be modeled by an exponential function. If you start with a base balance (the asymptote) and it grows exponentially, the total value will always be greater than that starting balance, just like the function \(h(x) = (1.4)^x + 5\) is always greater than 5. Knowing the domain and range helps you predict the possible values your account could reach over time.