For the function, supply a valid technology formula.
r(x) = 80

1 +
1
3.8
^4x

r(x) =
Use technology to compute the missing values in the table accurate to four significant digits.
x	−3	−2	−1	0	1	2	3
r(x)

To find the missing values of \( r(x) \) for the given function and table, we'll first rewrite the function for clarity:

\[
r(x) = 80 \left( 1 + \frac{1}{3.8^{4x}} \right)
\]

Now, we will compute the value of \( r(x) \) for each \( x \) in the table using this formula. Each result will be rounded to four significant digits.

### Computation of \( r(x) \):

1. **For \( x = -3 \):**
   \[
   r(-3) = 80 \left( 1 + \frac{1}{3.8^{4(-3)}} \right)
   \]

2. **For \( x = -2 \):**
   \[
   r(-2) = 80 \left( 1 + \frac{1}{3.8^{4(-2)}} \right)
   \]

3. **For \( x = -1 \):**
   \[
   r(-1) = 80 \left( 1 + \frac{1}{3.8^{4(-1)}} \right)
   \]

4. **For \( x = 0 \):**
   \[
   r(0) = 80 \left( 1 + \frac{1}{3.8^{4(0)}} \right)
   \]

5. **For \( x = 1 \):**
   \[
   r(1) = 80 \left( 1 + \frac{1}{3.8^{4(1)}} \right)
   \]

6. **For \( x = 2 \):**
   \[
   r(2) = 80 \left( 1 + \frac{1}{3.8^{4(2)}} \right)
   \]

7. **For \( x = 3 \):**
   \[
   r(3) = 80 \left( 1 + \frac{1}{3.8^{4(3)}} \right)
   \]

Let's compute these values now.Here are the computed values of \( r(x) \) rounded to four significant digits:

\[
\begin{array}{c|c}
x & r(x) \\
\hline
-3 & 7.253 \times 10^8 \\
-2 & 3.478 \times 10^6 \\
-1 & 1.676 \times 10^4 \\
0 & 160.0 \\
1 & 80.38 \\
2 & 80.00 \\
3 & 80.00 \\
\end{array}
\]

Would you like further details or explanations on these calculations? Here are some follow-up questions that might interest you:

1. How does changing the base of the exponential term affect the results?
2. What happens to \( r(x) \) as \( x \) approaches very large positive or negative values?
3. How would the function behave if we altered the coefficient 80 to another value?
4. Can this function be represented graphically, and what would the graph look like?
5. How does the function \( r(x) \) relate to real-world applications?

**Tip:** When working with exponential functions, pay close attention to the growth rates, especially when the exponent involves a negative sign, as it leads to decay rather than growth.

For the function, supply a valid technology formula. r(x) = 80 / (1 + 1 / 3.8^4x). Use technology to compute the missing values in the table accurate to four significant digits. x = {-3, -2, -1, 0, 1, 2, 3}.

Learn how to compute the values of the exponential function r(x) = 80(1 + 1/3.8^4x) for x = {-3, -2, -1, 0, 1, 2, 3}, with results accurate to four significant digits. This example covers exponential growth and decay concepts, suitable for Grades 10-12.

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