If `f:R to R` is defined by `f(x) = x^(2)-3x+2,` write `f{f(x)}`.

Explanation

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

### Step 1: Identify the function
We have the function defined as:
\[ f(x) = x^2 - 3x + 2 \]

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

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

### Step 4: Expand \( (x^2 - 3x + 2)^2 \)
Using the formula \( (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc \):
\[ (x^2 - 3x + 2)^2 = (x^2)^2 + (-3x)^2 + (2)^2 + 2(x^2)(-3x) + 2(-3x)(2) + 2(x^2)(2) \]
Calculating each term:
- \( (x^2)^2 = x^4 \)
- \( (-3x)^2 = 9x^2 \)
- \( (2)^2 = 4 \)
- \( 2(x^2)(-3x) = -6x^3 \)
- \( 2(-3x)(2) = -12x \)
- \( 2(x^2)(2) = 4x^2 \)

Combining these gives:
\[ x^4 - 6x^3 + (9x^2 + 4x^2) + 4 - 12x = x^4 - 6x^3 + 13x^2 - 12x + 4 \]

### Step 5: Substitute back into the equation
Now we substitute back into the equation for \( f(f(x)) \):
\[ f(f(x)) = (x^4 - 6x^3 + 13x^2 - 12x + 4) - 3(x^2 - 3x + 2) + 2 \]

### Step 6: Expand and simplify
Now we expand \( -3(x^2 - 3x + 2) \):
\[ -3(x^2 - 3x + 2) = -3x^2 + 9x - 6 \]
Now substituting this back:
\[ f(f(x)) = x^4 - 6x^3 + 13x^2 - 12x + 4 - 3x^2 + 9x - 6 + 2 \]
Combining like terms:
\[ f(f(x)) = x^4 - 6x^3 + (13x^2 - 3x^2) + (-12x + 9x) + (4 - 6 + 2) \]
This simplifies to:
\[ f(f(x)) = x^4 - 6x^3 + 10x^2 - 3x \]

### Final Result
Thus, the final expression for \( f(f(x)) \) is:
\[ f(f(x)) = x^4 - 6x^3 + 10x^2 - 3x \]