To transform \( f(x) + x^2 \) into \( f(2x + 1) \), we need to understand that this involves a combination of substituting a new expression into the function \( f \) and manipulating the original expression.
Here’s a step-by-step breakdown of the order of operations involved in this transformation:
| Step | Operation | Result | |------|--------------------------------|---------------------------------| | 1 | Start with | \( f(x) + x^2 \) | | 2 | Identify the substitution | Let \( x \) be replaced by \( 2x + 1 \) | | 3 | Apply substitution to \( f \) | \( f(2x + 1) + (2x + 1)^2 \) | | 4 | Simplify \( (2x + 1)^2 \) | \( = 4x^2 + 4x + 1 \) | | 5 | Combine results | \( f(2x + 1) + 4x^2 + 4x + 1 \) |
Thus, the transformation shows that if we start with \( f(x) + x^2 \), through the substitution and simplification, we can relate it to \( f(2x + 1) \).
Note that while we derive \( f(2x + 1) \) from the original function, the additional term \( 4x^2 + 4x + 1 \) represents the expanded form of \( (2x + 1)^2 \) which was added contrary to just obtaining \( f(2x + 1) \).
If you meant a more direct functional transformation that keeps the equality, it would depend on \( f \) and context. In symbolic transformation without knowing \( f \), you can't solely derive \( f(2x + 1) \) from \( f(x) + x^2 \) without further context or properties of \( f \).