Correctly show the order of operations that transforms f(x) = 3x

To transform the function \( f(x) = 3^x \) into \( -f(3x - 2) + 1 \), let's break down the steps in the order we would perform them. To create the table, we'll first write the new function and then show the sequential evaluations step by step.

New Function

• Start with: \( -f(3x - 2) + 1 \)

Steps

1. Evaluate: \( f(3x - 2) = 3^{(3x - 2)} \)
2. Apply the negative: \( -f(3x - 2) = -3^{(3x - 2)} \)
3. Add 1: \( -3^{(3x - 2)} + 1 \)

Now we can structure the evaluation for given \( x \) values in the table.

Table

For each row in the table, we will evaluate step by step, assuming the values of \( x \):

| x | STEP 1 | STEP 2 | STEP 3 | STEP 4 | STEP 5 | |---|-------------------------------|-----------------------------|---------------------------------|-----------------------------|---------------------------| | 1 | \( f(3 \cdot 1 - 2) = f(1) \) | \( 3^{1} = 3 \) | \( -3^{1} = -3 \) | \( -3 + 1 = -2 \) | \( -2 \) | | 2 | \( f(3 \cdot 2 - 2) = f(4) \) | \( 3^{4} = 81 \) | \( -81 \) | \( -81 + 1 = -80 \) | \( -80 \) | | 3 | \( f(3 \cdot 3 - 2) = f(7) \) | \( 3^{7} = 2187 \) | \( -2187 \) | \( -2187 + 1 = -2186 \) | \( -2186 \) |

Explanation of Steps:

• STEP 1: Calculate \( f(3x - 2) \) which involves substituting \( x \) into \( 3x - 2 \) to find the input for \( f \).
• STEP 2: Evaluate the function \( f \) itself which outputs \( 3^{(3x - 2)} \).
• STEP 3: Apply the negative sign to the evaluated result from STEP 2.
• STEP 4: Add 1 to the result obtained in STEP 3.
• STEP 5: This shows the final output after all transformations.

Feel free to modify or expand this as necessary for specific evaluation of different values for \( x \).

Correctly show the order of operations that transforms f(x) = 3x
to −f(3x−2)+1
.

Complete the table.

In rows 2-4, you will put the number you find after evaluating the expression for the value of x.

(8 points)
x
In the order of operations, what would you do first?

STEP 1: STEP 2: STEP 3: STEP 4: STEP 5:
1 _ _ _ _ _
2 _ _ _ _ _
3 _ _ _ _ _

To transform the function \( f(x) = 3x \) into \( -f(3x - 2) + 1 \), let's go through the steps of this transformation systematically.

Transformation Steps

1. Start with: \( -f(3x - 2) + 1 \)
2. Substitute \( f(x) = 3x \) into this expression:
   • \( f(3x - 2) = 3(3x - 2) \)
3. The expression now becomes:
   • \( -3(3x - 2) + 1 \)
4. Distribute the negative:
   • \( -9x + 6 + 1 \)
5. Final expression:
   • \( -9x + 7 \)

Evaluation Table

Now let’s create a table for evaluating this based on given \( x \) values. We will evaluate the expression for \( x = 1, 2, 3 \).

| x | STEP 1 | STEP 2 | STEP 3 | STEP 4 | STEP 5 | |---|---------------------------|---------------------------|--------------------------|----------------------------|------------| | 1 | \( -f(3 \cdot 1 - 2) + 1 \) | \( -f(1) + 1 \) | \( -3(1) + 1 = -3 + 1 \) | \( -3 + 1 = -2 \) | \( -2 \) | | 2 | \( -f(3 \cdot 2 - 2) + 1 \) | \( -f(4) + 1 \) | \( -3(4) + 1 = -12 + 1 \) | \( -12 + 1 = -11 \) | \( -11 \) | | 3 | \( -f(3 \cdot 3 - 2) + 1 \) | \( -f(7) + 1 \) | \( -3(7) + 1 = -21 + 1 \) | \( -21 + 1 = -20 \) | \( -20 \) |

Explanation of Steps:

• STEP 1: Calculate \( f(3x - 2) \) which involves substituting \( x \) into \( 3x - 2 \) to find the input for \( f \).
• STEP 2: Evaluate the function \( f \) itself which outputs \( 3(3x - 2) \).
• STEP 3: Apply the negative sign to the evaluated result.
• STEP 4: Add 1 to the result obtained in STEP 3.
• STEP 5: This shows the final output after all transformations.

This table now completes the steps for transforming and evaluating the function for \( x = 1, 2, 3 \).