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

Question

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

Complete the table.

You should write the new function first to help you fill out the table.

In the table in row 1, you should put the expression used to find the value of x in the correct order of operations. Progress by rewriting the expression in each subsequent step, applying an additional operation one at a time, until all 5 steps are complete.

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

Drag the steps into the chart

(5 points)

x   step 1           step 2               step 3                step 4            step 5 
1
2
3
answers
3x 
3x3^z
-(3x-2)+1
3
-3^(3z-2)
-3x+3
3x-1
3x-2
3^(3z-2)
(-3x+2)+1
-3^(3z-2)+1

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

GPT-4o mini · Answer

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 $.

Answer

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 _ _ _ _ _

GPT-4o mini · Answer

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 $.