Pregunta
After checking the bacteria population that was assigned to you, you remember that Casey had asked you to provide some analysis for the growth of their bacteria that they are hoping will speed up the production of an important material. Casey has asked you to take the function that models the population size, find the domain, range, and horizontal asymptote, and then plot the results. Casey specifically asked for these values related to the function that fits the data. They also wanted the input value of zero to represent the date that the data was collected. This makes negative input values represent previous days and positive values represent future predictions. After careful analysis, you determine that the function modeling the bacteria population is \( f(x)=4(2)^{x}+6 \). You next carefully find the domain and range in interval notation. To enter \( \infty \) when in text mode, type infinity. To enter \( \infty \) when in symbol mode, choose the \( \infty \) symbol. Domain: Range: asymptote: \( y= \)
Ask by Gibbs Mcfarlane. in the United States
Jan 30,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
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- **Domain:** All real numbers
- **Range:** Greater than 6
- **Asymptote:** \( y = 6 \)
Solución
Based on the function \( f(x) = 4(2)^x + 6 \), here are the requested analyses:
- **Domain:** \((-infinity, infinity)\)
The function is defined for all real numbers \( x \), meaning there are no restrictions on the input values.
- **Range:** \((6, infinity)\)
Since \( 4(2)^x \) is always positive for all real \( x \), adding 6 shifts the entire range upwards. Therefore, the population size will always be greater than 6.
- **Horizontal Asymptote:** \( y = 6 \)
As \( x \) approaches negative infinity, \( 4(2)^x \) approaches 0. Thus, the function approaches \( y = 6 \), which serves as the horizontal asymptote.
**Summary:**
- **Domain:** \((-infinity, infinity)\)
- **Range:** \((6, infinity)\)
- **Asymptote:** \( y = 6 \)
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Simplifique esta solución Extra Insights
The function given is \( f(x)=4(2)^{x}+6 \). Domain: The domain of this function is all real numbers, since there are no restrictions on \( x \). In interval notation, it's represented as \( (-\infty, \infty) \). Range: As \( x \) approaches negative infinity, \( f(x) \) approaches 6. As \( x \) approaches positive infinity, \( f(x) \) goes to infinity. Thus, the range in interval notation is \( (6, \infty) \). Asymptote: The horizontal asymptote of the function is \( y=6 \), meaning as time goes back indefinitely (negative \( x \)), the bacteria population will stabilize at this value.
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