To solve the problem of evaluating the exponential function [tex]\( f(x) = 1000(1.3)^x \)[/tex] at the given points, follow these steps for each part:
First, understand the structure of the exponential function:
[tex]\[ f(x) = 1000 \times (1.3)^x \][/tex]
Now, we'll evaluate the function at the specified points and round the results to two decimal places. Here are the detailed steps:
### Part A: Evaluate [tex]\( f(-10) \)[/tex]
1. Substitute [tex]\(-10\)[/tex] for [tex]\( x \)[/tex] in the function:
[tex]\[ f(-10) = 1000 \times (1.3)^{-10} \][/tex]
2. Calculate the result:
[tex]\[ f(-10) \approx 72.54 \][/tex]
So, [tex]\( f(-10) = 72.54 \)[/tex].
### Part B: Evaluate [tex]\( f(-5) \)[/tex]
1. Substitute [tex]\(-5\)[/tex] for [tex]\( x \)[/tex] in the function:
[tex]\[ f(-5) = 1000 \times (1.3)^{-5} \][/tex]
2. Calculate the result:
[tex]\[ f(-5) \approx 269.33 \][/tex]
So, [tex]\( f(-5) = 269.33 \)[/tex].
### Part C: Evaluate [tex]\( f(0) \)[/tex]
1. Substitute [tex]\(0\)[/tex] for [tex]\( x \)[/tex] in the function:
[tex]\[ f(0) = 1000 \times (1.3)^0 \][/tex]
2. Calculate the result (since any number to the power of 0 is 1):
[tex]\[ f(0) = 1000 \times 1 = 1000.00 \][/tex]
So, [tex]\( f(0) = 1000.00 \)[/tex].
### Part D: Evaluate [tex]\( f(5) \)[/tex]
1. Substitute [tex]\(5\)[/tex] for [tex]\( x \)[/tex] in the function:
[tex]\[ f(5) = 1000 \times (1.3)^5 \][/tex]
2. Calculate the result:
[tex]\[ f(5) \approx 3712.93 \][/tex]
So, [tex]\( f(5) = 3712.93 \)[/tex].
### Part E: Evaluate [tex]\( f(10) \)[/tex]
1. Substitute [tex]\(10\)[/tex] for [tex]\( x \)[/tex] in the function:
[tex]\[ f(10) = 1000 \times (1.3)^{10} \][/tex]
2. Calculate the result:
[tex]\[ f(10) \approx 13785.85 \][/tex]
So, [tex]\( f(10) = 13785.85 \)[/tex].
We can now fill in the table with the evaluated values:
[tex]\[
\begin{tabular}{|l|l|}
\hline \multicolumn{2}{|c|}{ Exponential Function Evaluation } \\
\hline Given the function \( f(x)=1000(1.3)^x \) evaluate each of the following. \\
Note: Round your answers to two decimal places as needed. \\
\hline A) Evaluate \( f(-10) \) & \( f(-10) = 72.54 \) \\
\hline B) Evaluate \( f(-5) \) & \( f(-5) = 269.33 \) \\
\hline C) Evaluate \( f(0) \) & \( f(0) = 1000.00 \) \\
\hline D) Evaluate \( f(5) \) & \( f(5) = 3712.93 \) \\
\hline E) Evaluate \( f(10) \) & \( f(10) = 13785.85 \) \\
\hline
\end{tabular}
\][/tex]
