If `f(x)=x^(2)-2x`, what is `f(2x+1)`?

Options

A. `4x+2`
B. `4x^(2)-1`
C. `4x^(2)+1`
D. `4x^(2)-4x-1`

Answer

To find \( f(2x + 1) \) given that \( f(x) = x^2 - 2x \), we will follow these steps:

### Step 1: Substitute \( 2x + 1 \) into the function \( f(x) \)
We start by substituting \( 2x + 1 \) in place of \( x \) in the function \( f(x) \).

\[ f(2x + 1) = (2x + 1)^2 - 2(2x + 1) \]

### Step 2: Expand \( (2x + 1)^2 \)
Next, we need to expand the square:

\[ (2x + 1)^2 = (2x)^2 + 2 \cdot (2x) \cdot (1) + (1)^2 = 4x^2 + 4x + 1 \]

### Step 3: Expand \( -2(2x + 1) \)
Now we expand the second part:

\[ -2(2x + 1) = -4x - 2 \]

### Step 4: Combine the results
We combine the results from Step 2 and Step 3:

\[ f(2x + 1) = (4x^2 + 4x + 1) + (-4x - 2) \]

### Step 5: Simplify the expression
Now we simplify the expression by combining like terms:

\[ f(2x + 1) = 4x^2 + 4x - 4x + 1 - 2 = 4x^2 - 1 \]

### Final Answer
Thus, the value of \( f(2x + 1) \) is:

\[ \boxed{4x^2 - 1} \]

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