Part AUse Estimation.The Function F ( X ) = 2 + 10 Log ⁡ 9 ( X + 2 F(x) = 2 + 10 \log_9(x + 2 F ( X ) = 2 + 10 Lo G 9 ( X + 2 ] Gives The Estimated Number Of Deer In A County, Where X X X Is The Number Of Years After 2012 And F ( X F(x F ( X ] Is The Deer Population In Hundreds. A Natural Resources

Mar 13, 2025 by ADMIN 302 views

Part A: Use Estimation to Solve the Function

Understanding the Function

The function $f(x) = 2 + 10 \log_9(x + 2)$ is used to estimate the number of deer in a county. The variable $x$ represents the number of years after 2012, and the function $f(x)$ gives the deer population in hundreds. This means that if we input the number of years after 2012, the function will output the estimated number of deer in hundreds.

Estimating the Deer Population

To estimate the deer population, we need to plug in the value of $x$ into the function. Let's say we want to estimate the deer population in 2020, which is 8 years after 2012. We can plug in $x = 8$ into the function:

$f(8) = 2 + 10 \log_9(8 + 2)$

Simplifying the Function

To simplify the function, we can start by evaluating the expression inside the logarithm:

$8 + 2 = 10$

So, the function becomes:

$f(8) = 2 + 10 \log_9(10)$

Using Logarithmic Properties

We can use the property of logarithms that states $\log_a(a) = 1$ to simplify the function further:

$\log_9(10) = \frac{\log(10)}{\log(9)}$

Since $\log(10) = 1$ and $\log(9) = 0.9542$ , we can substitute these values into the function:

$f(8) = 2 + 10 \cdot \frac{1}{0.9542}$

Evaluating the Function

Now, we can evaluate the function by multiplying $10$ by $\frac{1}{0.9542}$ :

$f(8) = 2 + 10.48$

So, the estimated number of deer in 2020 is $2 + 10.48 = 12.48$ hundred deer.

Interpreting the Results

The estimated number of deer in 2020 is $12.48$ hundred deer. This means that there are approximately $1248$ deer in the county.

Conclusion

In this section, we used estimation to solve the function $f(x) = 2 + 10 \log_9(x + 2)$ . We plugged in the value of $x = 8$ into the function and simplified it using logarithmic properties. We then evaluated the function to get an estimated number of deer in 2020.

Part B: Understanding the Graph of the Function

Understanding the Graph

The graph of the function $f(x) = 2 + 10 \log_9(x + 2)$ is a continuous curve that represents the estimated number of deer in a county. The graph has a horizontal asymptote at $y = 2$ , which means that as $x$ approaches infinity, the function approaches $2$ .

Identifying Key Features

The graph of the function has several key features that are important to understand:

Domain: The domain of the function is all real numbers greater than or equal to $-2$ . This means that the function is defined for all values of $x$ greater than or equal to $-2$ .
Range: The range of the function is all real numbers greater than or equal to $2$ . This means that the function can take on any value greater than or equal to $2$ .
Asymptotes: The graph has a horizontal asymptote at $y = 2$ and a vertical asymptote at $x = -2$ .

Interpreting the Graph

The graph of the function can be interpreted in several ways:

Estimated Deer Population: The graph represents the estimated number of deer in a county. The x-axis represents the number of years after 2012, and the y-axis represents the estimated number of deer in hundreds.
Growth Rate: The graph shows the growth rate of the deer population over time. The function is increasing as $x$ increases, which means that the deer population is growing over time.

Conclusion

In this section, we understood the graph of the function $f(x) = 2 + 10 \log_9(x + 2)$ . We identified key features of the graph, including the domain, range, and asymptotes. We also interpreted the graph in terms of the estimated deer population and growth rate.

Part C: Using the Function to Solve Real-World Problems

Understanding Real-World Applications

The function $f(x) = 2 + 10 \log_9(x + 2)$ has several real-world applications:

Wildlife Management: The function can be used to estimate the number of deer in a county, which is important for wildlife management.
Conservation: The function can be used to track changes in the deer population over time, which is important for conservation efforts.
Research: The function can be used to study the growth rate of the deer population, which is important for research purposes.

Solving Real-World Problems

To solve real-world problems using the function, we need to plug in the value of $x$ into the function and evaluate it. Let's say we want to estimate the deer population in 2025, which is 13 years after 2012. We can plug in $x = 13$ into the function:

$f(13) = 2 + 10 \log_9(13 + 2)$

Simplifying the Function

To simplify the function, we can start by evaluating the expression inside the logarithm:

$13 + 2 = 15$

So, the function becomes:

$f(13) = 2 + 10 \log_9(15)$

Using Logarithmic Properties

We can use the property of logarithms that states $\log_a(a) = 1$ to simplify the function further:

$\log_9(15) = \frac{\log(15)}{\log(9)}$

Since $\log(15) = 1.1761$ and $\log(9) = 0.9542$ , we can substitute these values into the function:

$f(13) = 2 + 10 \cdot \frac{1.1761}{0.9542}$

Evaluating the Function

Now, we can evaluate the function by multiplying $10$ by $\frac{1.1761}{0.9542}$ :

$f(13) = 2 + 12.32$

So, the estimated number of deer in 2025 is $2 + 12.32 = 14.32$ hundred deer.

Conclusion

In this section, we used the function $f(x) = 2 + 10 \log_9(x + 2)$ to solve real-world problems. We plugged in the value of $x = 13$ into the function and simplified it using logarithmic properties. We then evaluated the function to get an estimated number of deer in 2025.

Conclusion

In this article, we used estimation to solve the function $f(x) = 2 + 10 \log_9(x + 2)$ . We understood the graph of the function and identified key features, including the domain, range, and asymptotes. We also used the function to solve real-world problems, including estimating the deer population in 2025.
Part D: Q&A - Estimation and the Function

Q: What is the purpose of the function $f(x) = 2 + 10 \log_9(x + 2)$ ?

A: The purpose of the function is to estimate the number of deer in a county, where $x$ is the number of years after 2012 and $f(x)$ is the deer population in hundreds.

Q: How do I use the function to estimate the deer population?

A: To use the function to estimate the deer population, you need to plug in the value of $x$ into the function and evaluate it. For example, if you want to estimate the deer population in 2020, you would plug in $x = 8$ into the function.

Q: What is the domain of the function?

A: The domain of the function is all real numbers greater than or equal to $-2$ . This means that the function is defined for all values of $x$ greater than or equal to $-2$ .

Q: What is the range of the function?

A: The range of the function is all real numbers greater than or equal to $2$ . This means that the function can take on any value greater than or equal to $2$ .

Q: What is the horizontal asymptote of the function?

A: The horizontal asymptote of the function is $y = 2$ . This means that as $x$ approaches infinity, the function approaches $2$ .

Q: What is the vertical asymptote of the function?

A: The vertical asymptote of the function is $x = -2$ . This means that the function is undefined at $x = -2$ .

Q: How do I use the function to solve real-world problems?

A: To use the function to solve real-world problems, you need to plug in the value of $x$ into the function and evaluate it. For example, if you want to estimate the deer population in 2025, you would plug in $x = 13$ into the function.

Q: What are some real-world applications of the function?

A: Some real-world applications of the function include:

Wildlife management: The function can be used to estimate the number of deer in a county, which is important for wildlife management.
Conservation: The function can be used to track changes in the deer population over time, which is important for conservation efforts.
Research: The function can be used to study the growth rate of the deer population, which is important for research purposes.

Q: How do I simplify the function using logarithmic properties?

A: To simplify the function using logarithmic properties, you can use the property that states $\log_a(a) = 1$ . For example, if you have the function $f(x) = 2 + 10 \log_9(x + 2)$ , you can simplify it by using the property $\log_9(9) = 1$ .

Q: How do I evaluate the function using logarithmic properties?

A: To evaluate the function using logarithmic properties, you need to substitute the values of the logarithms into the function. For example, if you have the function $f(x) = 2 + 10 \log_9(x + 2)$ , you can evaluate it by substituting the values of the logarithms into the function.

Conclusion

In this article, we answered some common questions about the function $f(x) = 2 + 10 \log_9(x + 2)$ . We discussed the purpose of the function, how to use it to estimate the deer population, and some real-world applications of the function. We also discussed how to simplify and evaluate the function using logarithmic properties.