Show the table that correctly shows the order of operations that transforms f(x)+x^2 to f(2x +1)

5 answers

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 \( f \).

Table 1
x Step1 x^2 Step 2 2x^2 Step3 2x^2 +1
-1 1 2 3
0 0 0 1
2 4 8 9
Table 2
x Step1 x+1 Step2 (x+1)^2 Step3 2(x+1)^2
-1 0 0 0
0 1 1 2
2 3 9 18
Table3
x Step 1 2x Step 2 (2x)^2 Stpe3 2(x)^2+1
-1 -2 4 5
0 0 0 1
2 4 16 17
Table4
x Step1 2x Step2 2x+1 Step 3 (2x +1) ^2
-1 -2 -1 1
0 0 1 1
2 4 5 25

Based on the information you've provided, it looks like you have multiple tables showcasing various transformations of mathematical expressions involving \( x \). To clarify these transformations, I'll summarize each table based on their steps and purposes.

Table 1

This table demonstrates the transformation from \( x^2 \) through a series of steps.

| x | Step 1: \( x^2 \) | Step 2: \( 2x^2 \) | Step 3: \( 2x^2 + 1 \) | |-----|-------------------|---------------------|--------------------------| | -1 | 1 | 2 | 3 | | 0 | 0 | 0 | 1 | | 2 | 4 | 8 | 9 |

Table 2

This table shows the transformation starting from \( x + 1 \) to \( 2(x + 1)^2 \).

| x | Step 1: \( x + 1 \) | Step 2: \( (x + 1)^2 \) | Step 3: \( 2(x + 1)^2 \) | |-----|----------------------|-------------------------|--------------------------| | -1 | 0 | 0 | 0 | | 0 | 1 | 1 | 2 | | 2 | 3 | 9 | 18 |

Table 3

This table transforms \( 2x \) into \( 2(x^2) + 1 \).

| x | Step 1: \( 2x \) | Step 2: \( (2x)^2 \) | Step 3: \( 2(x)^2 + 1 \) | |-----|-------------------|-----------------------|---------------------------| | -1 | -2 | 4 | 5 | | 0 | 0 | 0 | 1 | | 2 | 4 | 16 | 17 |

Table 4

This table tracks the transformation from \( 2x \) to \( (2x + 1)^2 \).

| x | Step 1: \( 2x \) | Step 2: \( 2x + 1 \) | Step 3: \( (2x + 1)^2 \) | |-----|-------------------|-----------------------|---------------------------| | -1 | -2 | -1 | 1 | | 0 | 0 | 1 | 1 | | 2 | 4 | 5 | 25 |

Summary

  • Table 1 calculates the transformation starting from \( x^2 \).
  • Table 2 evaluates the expression \( (x + 1)^2 \) scaled by 2.
  • Table 3 focuses on \( 2x \) and relates it to \( 2(x)^2 + 1 \).
  • Table 4 computes \( (2x + 1)^2 \) after introducing the +1.

If the goal is to connect \( f(x) + x^2 \) to \( f(2x + 1) \) through a step/operation methodology like these tables, you may need a clearer function \( f \) context.

Each table illustrates transforming specific expressions, but they do not necessarily create a direct transformation from \( f(x) + x^2 \) to \( f(2x + 1) \) without further information about function \( f \). If you would like to analyze any specific transformations or resolve functions in a more focused context, please let me know!

Which table shows the correct order of operations that transforms f(x)=x^2 to f(2x + 1)

To transform \( f(x) = x^2 \) into \( f(2x + 1) \), we follow these steps:

  1. Replace \( x \) in \( f(x) \) with \( 2x + 1 \).
  2. Calculate the resulting expression.

Let's analyze the tables you provided to identify which one reflects this transformation.

Table Analysis

  • Table 1 involves the transformations of \( x \) through its square \( x^2 \), then scaling it and adding 1. This does not match the desired transformation.

  • Table 2 similarly evaluates \( (x + 1)^2 \) and then multiplies by 2. Again, this transformation does not match what we are looking for.

  • Table 3 shows \( 2x \) then to \( (2x)^2 \) and finally attempts to formulate \( 2(x)^2 + 1 \). This is not the process needed for transforming \( x^2 \) to \( f(2x + 1) \).

  • Table 4 computes the transformation by first calculating \( 2x \), then \( 2x + 1 \), and finally it computes \( (2x + 1)^2 \).

The critical step is the computation of \( (2x + 1)^2 \).

Thus, since the function \( f(x) = x^2 \) becomes \( f(2x + 1) = (2x + 1)^2 \) when substituting \( x \) with \( 2x + 1 \), Table 4 reflects the correct order of operations that shows how to go from \( f(x) = x^2 \) to \( f(2x + 1) \):

Table 4 - Correct Steps

| x | Step 1: \( 2x \) | Step 2: \( 2x + 1 \) | Step 3: \( (2x + 1)^2 \) | |-----|-------------------|-----------------------|---------------------------| | -1 | -2 | -1 | 1 | | 0 | 0 | 1 | 1 | | 2 | 4 | 5 | 25 |

In summary, Table 4 correctly demonstrates the transformation of \( f(x) = x^2 \) to \( f(2x + 1) \).