Question 1

To transform the function \( f(x) = 3x \) into \( -f(3x - 2) + 1 \), we can break it down into a series of operations. Let's go through the transformations step by step:

1. Start with \( f(x) = 3x \).
2. Substitute \( 3x - 2 \) into \( f \): This gives us \( f(3x - 2) = 3(3x - 2) \).
3. Multiply by -1: We then take the negative of the function, yielding \( -f(3x - 2) = -3(3x - 2) \).
4. Add 1: Finally, we add 1 to the function, obtaining \( -3(3x - 2) + 1 \).

Now let's fill out the table based on these operations:

| Step | Expression | Value of Expression for \( x \) | |--------------------------------------------------|------------------------------------|--------------------------------------| | 1. Start with the original function | \( f(x) = 3x \) | \( 3x \) | | 2. Substitute \( 3x - 2 \) | \( f(3x - 2) = 3(3x - 2) \) | \( 3(3x - 2) \) | | 3. Apply negative function | \( -f(3x - 2) = -3(3x - 2) \) | \( -3(3x - 2) \) | | 4. Add 1 to the function | \( -3(3x - 2) + 1 \) | \( -3(3x - 2) + 1 \) | | Final expression | Complete transformation: | \( -3(3x - 2) + 1 \) |

To summarize:

• Step 1 starts with \( f(x) = 3x \)
• Step 2 replaces \( x \) with \( 3x - 2 \)
• Step 3 negates the value of the function
• Step 4 adds 1 to the result to complete the transformation into \( -f(3x - 2) + 1 \)

Please evaluate \( 3x \), \( f(3x - 2) \), and \( -f(3x - 2) + 1 \) for specific values of \( x \) if needed.

Apologies for the confusion! Let’s break down the transformation into 5 distinct steps and fill out the table accordingly.

To transform \( f(x) = 3x \) into \( -f(3x - 2) + 1 \) using the correct order of operations through 5 steps:

Transformation Steps:

1. Start with \( f(x) = 3x \).
2. Substitute \( 3x - 2 \) for \( x \): Calculate \( f(3x - 2) = 3(3x - 2) \).
3. Calculate \( 3(3x - 2) \): This results in \( 9x - 6 \).
4. Negate the entire function: This gives \( - (9x - 6) = -9x + 6 \).
5. Add 1: Finally, you get \( -9x + 6 + 1 = -9x + 7 \).

Now, let's fill out the table:

| Step | Expression | Value of Expression for \( x \) | |--------------------------------------------------|------------------------------------|--------------------------------------| | 1. Start with the original function | \( 3x \) | \( 3x \) | | 2. Substitute \( 3x - 2 \) | \( f(3x - 2) = 3(3x - 2) \) | \( 3(3x - 2) \) | | 3. Evaluate \( 3(3x - 2) \) | \( 9x - 6 \) | \( 9x - 6 \) | | 4. Negate the function | \( - (9x - 6) \) | \( -9x + 6 \) | | 5. Add 1 | \( -9x + 6 + 1 \) | \( -9x + 7 \) |

This structure clearly outlines each step as we transform \( f(x) = 3x \) into \( -f(3x - 2) + 1 \). Each line in the table represents a distinct operation within the transformation process, leading to the final expression.