Understanding the Problem: Finding f(x) from f(x+1)

The question asks us to find the expression for a function f(x) given the expression for f(x + 1). We are provided that \(f(x + 1) = x^2 - 3x + 2\). This means that whatever is inside the parentheses of f is being processed according to the rule \((\text{input} - 1)^2 - 3(\text{input} - 1) + 2\). We need to find the rule when the input is simply 'x'.

Step-by-Step Solution for Function Evaluation

To find f(x) from f(x+1), we can use a substitution method. The goal is to replace the expression \((x+1)\) with a single variable, say \(y\), and then express the original variable \(x\) in terms of this new variable \(y\). Once we have an expression for f(y), we can simply replace \(y\) with \(x\) to get f(x).

1. Set up the substitution: Let \(y = x + 1\).
2. Express the original variable in terms of the new variable: From \(y = x + 1\), we can solve for \(x\):

   \(x = y - 1\)

3. Substitute into the given equation: The given equation is \(f(x + 1) = x^2 - 3x + 2\). Since we let \(y = x+1\), the left side becomes \(f(y)\). On the right side, we replace every instance of \(x\) with \((y - 1)\).

   \(f(y) = (y - 1)^2 - 3(y - 1) + 2\)

4. Expand and simplify the expression: Now, we simplify the right side of the equation.
   • Expand \((y - 1)^2\): This is a perfect square trinomial, \((a-b)^2 = a^2 - 2ab + b^2\). So, \((y - 1)^2 = y^2 - 2(y)(1) + 1^2 = y^2 - 2y + 1\).
   • Expand \(-3(y - 1)\): Distribute the \(-3\): \(-3(y - 1) = -3y - (-3)(1) = -3y + 3\).

   Substitute these back into the equation for f(y):

   \(f(y) = (y^2 - 2y + 1) + (-3y + 3) + 2\)

   Combine like terms:

   \(f(y) = y^2 + (-2y - 3y) + (1 + 3 + 2)\)

   \(f(y) = y^2 - 5y + 6\)

5. Replace the variable to find f(x): We found the expression for f(y). To find f(x), we simply replace the variable \(y\) with \(x\).

   \(f(x) = x^2 - 5x + 6\)

Verifying the Solution

We can quickly check our answer by substituting \((x+1)\) back into our derived function \(f(x) = x^2 - 5x + 6\). If we did the calculation correctly, \(f(x+1)\) should equal the original expression \(x^2 - 3x + 2\).

Let's calculate \(f(x+1)\) using \(f(x) = x^2 - 5x + 6\):

\(f(x+1) = (x+1)^2 - 5(x+1) + 6\)

Expand the terms:

• \((x+1)^2 = x^2 + 2x + 1\)
• \(-5(x+1) = -5x - 5\)

Substitute back and simplify:

\(f(x+1) = (x^2 + 2x + 1) + (-5x - 5) + 6\)

\(f(x+1) = x^2 + 2x - 5x + 1 - 5 + 6\)

\(f(x+1) = x^2 - 3x + 2\)

This matches the original given expression for \(f(x+1)\), confirming our derived function \(f(x)\) is correct.

Comparing with Options

Our calculated expression for f(x) is \(x^2 - 5x + 6\). Let's look at the given options:

Comparing our result \(x^2 - 5x + 6\) with the options, we see it matches Option 2.

Revision Table: Function Evaluation

Additional Information: Transforming Function Arguments

Problems like this involve understanding how changing the input of a function affects the output expression. When you have \(f(g(x))\) and want to find \(f(x)\), you essentially treat \(g(x)\) as a single variable (let's call it \(y\)), solve for \(x\) in terms of \(y\), and substitute that expression for \(x\) into the given formula for \(f(g(x))\). Finally, replace \(y\) with \(x\).

For example, if you were given \(f(2x) = 4x^2 + 1\), to find \(f(x)\):

1. Let \(y = 2x\).
2. Solve for \(x\): \(x = y/2\).
3. Substitute into the expression: \(f(y) = 4(y/2)^2 + 1\).
4. Simplify: \(f(y) = 4(y^2/4) + 1 = y^2 + 1\).
5. Replace \(y\) with \(x\): \(f(x) = x^2 + 1\).

This technique is broadly applicable when the argument of the function is an expression involving the variable, and you need to find the function rule for a simpler argument like just \(x\).