To transform \( f(x) \) based on the expression \( f(3x - 2) + 1 \), we want to apply the order of operations in a step-by-step manner. Let’s label this transformation and break it down accordingly.
Starting with the expression \( 3x - 2 \):
- Step 1: We start with the expression \( 3x - 2 \).
- Step 2: We evaluate \( 3x - 2 \) first.
- Step 3: Then, we apply \( 3^{(3x - 2)} \).
- Step 4: Now, we apply the negative, \( -3^{(3x - 2)} \).
- Step 5: Finally, we add 1, resulting in \( -3^{(3x - 2)} + 1 \).
Now, let's fill in the table with these computations:
| Step | Expression | Result | |-------|-----------------------------------------------------|---------------| | Step 1| Start with \( 3x - 2 \) | - | | Step 2| Evaluate \( 3x - 2 \) | \( 3x - 2 \) | | Step 3| Calculate \( 3^{(3x - 2)} \) | \( 3^{(3x - 2)} \) | | Step 4| Apply negative: \( -3^{(3x - 2)} \) | \( -3^{(3x - 2)} \) | | Step 5| Final step: \( -3^{(3x - 2)} + 1 \) | \( -3^{(3x - 2)} + 1 \) |
Now, let's put in the values:
-
For \( x = 1 \):
- Step 1: \( 3(1) - 2 = 1 \)
- Step 2: \( 3^{1} = 3 \)
- Step 3: \( -3 = -3 \)
- Step 4: \( -3 + 1 = -2 \)
-
For \( x = 2 \):
- Step 1: \( 3(2) - 2 = 4 \)
- Step 2: \( 3^{4} = 81 \)
- Step 3: \( -81 = -81 \)
- Step 4: \( -81 + 1 = -80 \)
-
For \( x = 3 \):
- Step 1: \( 3(3) - 2 = 7 \)
- Step 2: \( 3^{7} = 2187 \)
- Step 3: \( -2187 = -2187 \)
- Step 4: \( -2187 + 1 = -2186 \)
Make sure to verify each computation at each step to ensure accuracy based on the transformation of the function.