If `f:RrarrR` defined by `f(x)=x^(2)-2x+3`, then find `f(f(x))`.

Answer

To find \( f(f(x)) \) for the function \( f(x) = x^2 - 2x + 3 \), we will follow these steps:

### Step 1: Write down the function
We start with the function:
\[ f(x) = x^2 - 2x + 3 \]

### Step 2: Substitute \( f(x) \) into itself
To find \( f(f(x)) \), we need to substitute \( f(x) \) into the function \( f \):
\[ f(f(x)) = f(x^2 - 2x + 3) \]

### Step 3: Replace \( x \) in \( f(x) \) with \( x^2 - 2x + 3 \)
Now we will replace \( x \) in the original function with \( x^2 - 2x + 3 \):
\[ f(f(x)) = (x^2 - 2x + 3)^2 - 2(x^2 - 2x + 3) + 3 \]

### Step 4: Expand \( (x^2 - 2x + 3)^2 \)
We need to expand \( (x^2 - 2x + 3)^2 \):
\[ (x^2 - 2x + 3)^2 = x^4 - 4x^3 + 6x^2 - 4x + 9 \]

### Step 5: Expand \( -2(x^2 - 2x + 3) \)
Next, we expand \( -2(x^2 - 2x + 3) \):
\[ -2(x^2 - 2x + 3) = -2x^2 + 4x - 6 \]

### Step 6: Combine all parts
Now we combine all the parts together:
\[ f(f(x)) = (x^4 - 4x^3 + 6x^2 - 4x + 9) + (-2x^2 + 4x - 6) + 3 \]

### Step 7: Simplify the expression
Now, we simplify the expression:
\[ f(f(x)) = x^4 - 4x^3 + (6x^2 - 2x^2) + (-4x + 4x) + (9 - 6 + 3) \]
\[ = x^4 - 4x^3 + 4x^2 + 6 \]

### Final Answer
Thus, the final answer is:
\[ f(f(x)) = x^4 - 4x^3 + 4x^2 + 6 \]

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