Question
Correctly show the order of operations that transforms f(x)+3^x to -f(3x-2)+1
Answers
GPT-4o mini
To transform \( f(x) + 3^x \) into \(-f(3x - 2) + 1\), we can break down this transformation step by step using the order of operations and the appropriate function manipulations.
1. **Starting Expression**:
\[
f(x) + 3^x
\]
2. **Substituting into the Function**:
To relate it to \( -f(3x - 2) \), we need to investigate how we can manipulate \( f(x) \). The manipulation suggests we might be transforming the input of the function \( f \).
3. **Applying a Transformation**:
Let's consider the term \( f(3x - 2) \). We replace \( f(x) \) by its input form:
\[
f(x) \rightarrow f(3x - 2)
\]
4. **Negating the Function**:
We now need to apply the negative sign and add \( 1 \):
\[
-f(3x - 2) + 1
\]
5. **Consistency Check**:
To ensure that the transformation is valid within the functions we are dealing with, notice:
- The function is transformed from \( f(x) \) to \( f(3x - 2) \).
- The output is negated, indicating a reflection about the horizontal axis.
- Finally, \( +1 \) is added, moving the entire function upwards by 1 unit.
Hence, the transformation from \( f(x) + 3^x \) to \( -f(3x - 2) + 1 \) can be made clear with the steps involving function transformations and basic operations.
### Summary of Steps:
1. Start with \( f(x) + 3^x \).
2. Consider the transformation \( f(x) \to f(3x - 2) \).
3. Apply negation to the function: \( -f(3x - 2) \).
4. Add 1 to the result.
Therefore, the ordered operations result in:
\[
f(x) + 3^x \rightarrow -f(3x - 2) + 1
\]
1. **Starting Expression**:
\[
f(x) + 3^x
\]
2. **Substituting into the Function**:
To relate it to \( -f(3x - 2) \), we need to investigate how we can manipulate \( f(x) \). The manipulation suggests we might be transforming the input of the function \( f \).
3. **Applying a Transformation**:
Let's consider the term \( f(3x - 2) \). We replace \( f(x) \) by its input form:
\[
f(x) \rightarrow f(3x - 2)
\]
4. **Negating the Function**:
We now need to apply the negative sign and add \( 1 \):
\[
-f(3x - 2) + 1
\]
5. **Consistency Check**:
To ensure that the transformation is valid within the functions we are dealing with, notice:
- The function is transformed from \( f(x) \) to \( f(3x - 2) \).
- The output is negated, indicating a reflection about the horizontal axis.
- Finally, \( +1 \) is added, moving the entire function upwards by 1 unit.
Hence, the transformation from \( f(x) + 3^x \) to \( -f(3x - 2) + 1 \) can be made clear with the steps involving function transformations and basic operations.
### Summary of Steps:
1. Start with \( f(x) + 3^x \).
2. Consider the transformation \( f(x) \to f(3x - 2) \).
3. Apply negation to the function: \( -f(3x - 2) \).
4. Add 1 to the result.
Therefore, the ordered operations result in:
\[
f(x) + 3^x \rightarrow -f(3x - 2) + 1
\]