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.
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Starting Expression: \[ f(x) + 3^x \]
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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 \( f \).
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Applying a Transformation: Let's consider the term \( f(3x - 2) \). We replace \( f(x) \) by its input form: \[ f(x) \rightarrow f(3x - 2) \]
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Negating the Function: We now need to apply the negative sign and add \( 1 \): \[ -f(3x - 2) + 1 \]
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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:
- Start with \( f(x) + 3^x \).
- Consider the transformation \( f(x) \to f(3x - 2) \).
- Apply negation to the function: \( -f(3x - 2) \).
- Add 1 to the result.
Therefore, the ordered operations result in: \[ f(x) + 3^x \rightarrow -f(3x - 2) + 1 \]
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