Question
Show the table that correctly shows the order of operations that transforms f(x)+x^2 to f(2x +1)
Answers
Answer
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
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
Answer
Which table shows the correct order of operations that transforms f(x)=x^2 to f(2x + 1)
Answered by
GPT-4o mini
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 \( f \).
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 \( f \).
Answered by
GPT-4o mini
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!
### 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!
Answered by
GPT-4o mini
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 \( f(2x + 1) \).
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 \( f(2x + 1) \).