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

Table
x. Step 1 Step 2. Step 3. Step 4. Step5
_______ ____. ____. _____. ___
1 _____. ____. _____. _____. _____

2. _______. ____. ____. _____. ____

3. _______ _____. _____. _____. ____

Here are the options
3^(3x-2) -3x+3 3. 3x 3x-2
-3^(3x-2) +1 3x-1 -(3x-2)+1 3*3^x
(-3x+2)+1 -3^(3x-2)

5 answers

To transform \( f(x) = 3x \) into \( -f(3x - 2) + 1 \), we will break it down into steps while filling in the provided table.

We start with \( f(x) = 3x \) and want to calculate \( -f(3x - 2) + 1 \).

Step 1: Evaluate \( f(3x - 2) \) \[ f(3x - 2) = 3(3x - 2) = 9x - 6 \]

Step 2: Apply the negative \[ -f(3x - 2) = -(9x - 6) = -9x + 6 \]

Step 3: Add 1 \[ -f(3x - 2) + 1 = -9x + 6 + 1 = -9x + 7 \]

Now, we'll fill out the table with the steps of transformation.

Table

| x | Step 1 | Step 2 | Step 3 | Step 4 | Step 5 | |-------|--------------|-------------------|----------------|---------------------|----------------| | 1 | \( 3(3x - 2) \) | \( -9x + 6 \) | \( -9x + 7 \) | | | | 2 | \( 3(3x - 2) \) | \( -9x + 6 \) | \( -9x + 7 \) | | | | 3 | \( 3(3x - 2) \) | \( -9x + 6 \) | \( -9x + 7 \) | | |

Since none of the options exactly match our calculations or steps, here’s how we can fit the closest options to the steps:

Final Options

  • Step 1: \( 3(3x - 2) \) could be represented as \( 3^{(3x - 2)} \).
  • Step 2: Represents negative \( 9x + 6 \) could correspond to \( -3(3x - 2) + 1 \) or \( -9x + 6 \) (fitting into the options).
  • Step 3: Taking \( -f(3x - 2) + 1 \) would result in \( -9x + 7 \) or correct fits.

To summarize, the correct order of operations based on the transformation is:

  1. Calculate \( f(3x - 2) = 3(3x - 2) \)
  2. Apply negative: \( -(3(3x - 2)) + 1 \)
  3. Final result: calculate \( -(9x - 6) + 1 \).

Given the nature of the options, it seems a more precise interpretation of the options should be determined since not all completely match the pathways created based on the transformations.

So there is supposed to be 5 steps what are the other 2

My apologies for the confusion. Let's outline the complete transformation process step-by-step to clearly indicate the five steps in the transformation from \( f(x) = 3x \) to \( -f(3x - 2) + 1 \).

Steps to Transform \( f(x) = 3x \) to \( -f(3x - 2) + 1 \):

  1. Step 1: Start with the function \[ f(x) = 3x \]

  2. Step 2: Substitute \( 3x - 2 \) into the function \[ f(3x - 2) = 3(3x - 2) = 9x - 6 \]

  3. Step 3: Apply the negative \[ -f(3x - 2) = -(9x - 6) = -9x + 6 \]

  4. Step 4: Add 1 \[ -f(3x - 2) + 1 = -9x + 6 + 1 = -9x + 7 \]

  5. Step 5: Final form \[ -f(3x - 2) + 1 = -9x + 7 \]

Filling in the Table:

Now we will fill in the table with these steps while incorporating the options given. In relation to the options you provided, there are suitable representations we can match with each step in the transformation.

Table

| x | Step 1 | Step 2 | Step 3 | Step 4 | Step 5 | |---|----------------------|----------------------|---------------------|--------------------------|---------------------| | 1 | \( 3x \) | \( 3(3x-2) = 9x-6 \) | \( - (9x-6) = -9x + 6 \) | \( -9x + 6 + 1 = -9x + 7 \) | Final: \( -9x + 7 \) | | 2 | \( 3x \) | \( 3(3x-2) = 9x-6 \) | \( - (9x-6) = -9x + 6 \) | \( -9x + 6 + 1 = -9x + 7 \) | Final: \( -9x + 7 \) | | 3 | \( 3x \) | \( 3(3x-2) = 9x-6 \) | \( - (9x-6) = -9x + 6 \) | \( -9x + 6 + 1 = -9x + 7 \) | Final: \( -9x + 7 \) |

Matching with Given Options:

To align these steps with the available options, we can utilize:

  • Step 1: Rather than writing \( 3x \) directly, we refer to it as \( 3 \cdot x \) or even as starting from \( 3^{(3x-2)} \) in terms of just continuing from the original function leading into finding \( f(3x - 2) \).
  • Step 2: Corresponds ideally to calculate or represent \( 3(3x - 2) \).
  • Step 3: The representation of this can track with transformation actions resulting in \( -f(3x-2) + 1 \).
  • Step 4: Combining representations similar to \( -(3x - 2) + 1 \).

Thus, while examining the transformations step-by-step provides a clearer calculation route, it enables laying out options continuously.

To sum up:

The complete five-step transformation results from carefully evaluating each transition provided in order, accurately reflecting the nature of function transformations.

Ok get the first 5 steps but what about the 5 that goes with
1
2
3

I apologize for the misunderstanding in my previous messages. Let's clarify the five steps in detail and also align them with the intended uses of the specific examples you've provided in your table format while ensuring we have a complete understanding of the transformation process from \( f(x) = 3x \) to \( -f(3x - 2) + 1 \).

Transformation Steps:

  1. Step 1: Begin with \( f(x) \) \[ f(x) = 3x \]

  2. Step 2: Substitute \( 3x - 2 \) into the function \[ f(3x - 2) = 3(3x - 2) = 9x - 6 \]

  3. Step 3: Apply the negative to the function \[ -f(3x - 2) = -(9x - 6) = -9x + 6 \]

  4. Step 4: Add 1 to the result \[ -f(3x - 2) + 1 = -9x + 6 + 1 = -9x + 7 \]

  5. Step 5: Final expression \[ -f(3x - 2) + 1 = -9x + 7 \]

Filling in the Table Corresponding to Given Options:

Now let's match this process up with the table format you've provided using three rows, where I will assume each row corresponds to evaluating at different values of \( x \).

Table Format Filled with Options:

I'll use the provided options in the context of the operations we applied.

| \( x \) | Step 1 | Step 2 | Step 3 | Step 4 | Step 5 | |---------|-----------------------|------------------------|----------------------|-------------------------|---------------------| | 1 | \( f(x) = 3x \) | \( f(3(1) - 2) = f(1) = 3(1) = 3 \) | \( -3 \) | \( -3 + 1 = -2 \) | Final = \( -2 \) | | 2 | \( f(x) = 3(2) = 6 \) | \( f(3(2) - 2) = f(4) = 3(4) = 12 \) | \( -12 \) | \( -12 + 1 = -11 \) | Final = \( -11 \) | | 3 | \( f(x) = 3(3) = 9 \) | \( f(3(3) - 2) = f(7) = 3(7) = 21 \) | \( -21 \) | \( -21 + 1 = -20 \) | Final = \( -20 \) |

Explanation of Corresponding Steps:

  • Step 1 represents the original function.
  • Step 2 reflects calculating \( f(3x - 2) \) for specific values of \( x \).
  • Step 3 shows applying the negative.
  • Step 4 demonstrates the addition of 1.
  • Step 5 gives the final result after these transformations occur.

Matching the Options:

From your earlier options provided, here's how they would align roughly in correspondence to the operations:

  • Step 1 could represent: \( 3x \)
  • Step 2 could represent: \( 3(3x - 2) \) or \( 3x - 2 \)
  • Step 3 could represent: \( -(9x - 6) \)
  • Step 4 could represent: \( -9x + 7 \)

I hope this provides a clearer table context combined with specified option matches! If further clarification is required, please let me know.

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