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

To transform \( f(x) = 3x \) into \( -f(3x-2) + 1 \), we can break this transformation down into a series of steps by progressively applying operations. Here’s how we can fill out the table, showcasing the expression at each step and ultimately arriving at the final transformation.

Transformation Steps

| Step | Expression | Operation | |------|--------------------------------|--------------------------------------| | 1 | \( f(x) = 3x \) | Start with the original function. | | 2 | \( f(3x-2) = 3(3x-2) \) | Substitute \( 3x-2 \) into \( f \). | | 3 | \( 3(3x-2) = 9x - 6 \) | Apply multiplication. | | 4 | \( -f(3x-2) = -(9x - 6) \) | Apply the negative sign. | | 5 | \( -f(3x-2) + 1 = -(9x - 6) + 1 \) | Add 1 to the expression. | | 6 | \( -9x + 6 + 1 \) | Simplify the expression. | | 7 | \( -9x + 7 \) | Combine like terms. |

Final Result

The transformed expression from \( f(x) = 3x \) to \( -f(3x-2) + 1 \) is \( -9x + 7 \).

Conclusion

In the final transformed expression, after performing all the operations as outlined in the steps above, we arrive at \( -9x + 7 \). Each sequential transformation builds upon the previous step, illustrating the correct order of operations clearly. If you would like me to substitute a particular value for \( x \) and evaluate the final expression, please provide the value of \( x \).

Let's complete the table according to the steps needed to transform \( f(x) = 3x \) into \( -f(3x-2) + 1 \), filling out each corresponding step of the transformation.

Table Completion

| Step | Expression | Operation | |------|-------------------------------------|---------------------------------------| | 1 | \( f(x) = 3x \) | Start with the original function. | | 2 | \( f(3x-2) = 3(3x-2) \) | Substitute \( 3x-2 \) into \( f \). | | 3 | \( 3(3x-2) = 9x - 6 \) | Apply multiplication. | | 4 | \( -f(3x-2) = -(9x - 6) \) | Apply the negative sign. | | 5 | \( -f(3x-2) + 1 = -(9x - 6) + 1 \) | Add 1 to the expression. |

Now, let's summarize the transformations in the specific format you've requested.

Step Format

| STEP 1: | STEP 2: | STEP 3: | STEP 4: | STEP 5: | |---------------|--------------------|----------------|---------------------|----------------------------| | \( f(x) = 3x \) | \( f(3x-2) = 3(3x-2) \) | \( 3(3x-2) = 9x - 6 \) | \( -f(3x-2) = -(9x-6) \) | \( -f(3x-2) + 1 = -(9x - 6) + 1 \) |

Summary of Each Step:

1. Step 1: Start with the original function definition \( f(x) = 3x \).
2. Step 2: Evaluate \( f(3x-2) \) by substituting \( 3x-2 \) into \( f \) to get \( 3(3x-2) \).
3. Step 3: Calculate \( 3(3x-2) \) to arrive at \( 9x - 6 \).
4. Step 4: Apply the negative sign to the entire \( f(3x-2) \) to get \( -(9x-6) \).
5. Step 5: Finally, add 1 to the result from Step 4, yielding \( -(9x - 6) + 1 \).

Feel free to let me know if you want to evaluate this expression for a specific value of \( x \) or if you have any other questions!