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Problem 10

# aCompute the missing values in the following table and supply a valid technology formula for the given function: HINT [See Quick Examples on page 633.] $$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & & \\ \hline \end{array}$$ $$r(x)=2^{-x}+1$$

Expert verified
The missing values in the table are obtained by evaluating the function $$r(x) = 2^{-x} + 1$$ at the given x values: $$r(-3) = 9$$, $$r(-2) = 5$$, $$r(-1) = 3$$, $$r(0) = 2$$, $$r(1) = \frac{3}{2}$$, $$r(2) = \frac{5}{4}$$, and $$r(3) = \frac{9}{8}$$. The completed table is: $$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 9 & 5 & 3 & 2 & \frac{3}{2} & \frac{5}{4} & \frac{9}{8} \\ \hline \end{array}$$
See the step by step solution

## Step 1: Understand the function

We have the function r(x) = 2^{-x} + 1. We will evaluate this function for each x value given in the table and find the corresponding function value f(x).

## Step 2: Evaluate the function at x = -3

First, we plug in x = -3 into the function r(x): $$r(-3) = 2^{-(-3)} + 1 = 2^3 + 1 = 8 + 1 = 9$$

## Step 3: Evaluate the function at x = -2

Next, we plug in x = -2 into the function r(x): $$r(-2) = 2^{-(-2)} + 1 = 2^2 + 1 = 4 + 1 = 5$$

## Step 4: Evaluate the function at x = -1

Then, we plug in x = -1 into the function r(x): $$r(-1) = 2^{-(-1)} + 1 = 2^1 + 1 = 2 + 1 = 3$$

## Step 5: Evaluate the function at x = 0

Now, we plug in x = 0 into the function r(x): $$r(0) = 2^{-(0)} + 1 = 2^0 + 1 = 1 + 1 = 2$$

## Step 6: Evaluate the function at x = 1

Afterward, we plug in x = 1 into the function r(x): $$r(1) = 2^{-(1)} + 1 = 2^{-1} + 1 = \frac{1}{2} + 1 = \frac{3}{2}$$

## Step 7: Evaluate the function at x = 2

Next, we plug in x = 2 into the function r(x): $$r(2) = 2^{-(2)} + 1 = 2^{-2} + 1 = \frac{1}{4} + 1 = \frac{5}{4}$$

## Step 8: Evaluate the function at x = 3

Lastly, we plug in x = 3 into the function r(x): $$r(3) = 2^{-(3)} + 1 = 2^{-3} + 1 = \frac{1}{8} + 1 = \frac{9}{8}$$

## Step 9: Complete the table

Now we can fill in the missing values in the table: $$\begin{array}{|c|c|c|c|c|c|c|c|} \hline \boldsymbol{x} & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & 9 & 5 & 3 & 2 & \frac{3}{2} & \frac{5}{4} & \frac{9}{8} \\ \hline \end{array}$$

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