If `f : Rto R` is defined as `f (x) = x^(2) -3x +4 ` for all `x in R,` then `f ^(-1) (2)` is equal to

Options

A. `{1,2}`
B. `(1,2)`
C. `[1,2]`
D. none of these

Answer

To find \( f^{-1}(2) \) for the function \( f(x) = x^2 - 3x + 4 \), we need to follow these steps:

### Step 1: Set the function equal to 2
We start by setting the function equal to 2:
\[ f(x) = 2 \]
This gives us the equation:
\[ x^2 - 3x + 4 = 2 \]

### Step 2: Rearrange the equation
Next, we rearrange the equation to bring all terms to one side:
\[ x^2 - 3x + 4 - 2 = 0 \]
This simplifies to:
\[ x^2 - 3x + 2 = 0 \]

### Step 3: Factor the quadratic equation
Now we need to factor the quadratic equation:
\[ x^2 - 3x + 2 = (x - 1)(x - 2) = 0 \]

### Step 4: Solve for x
Setting each factor equal to zero gives us the solutions:
\[ x - 1 = 0 \quad \Rightarrow \quad x = 1 \]
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]

### Step 5: Conclusion
Thus, the values of \( x \) that satisfy \( f(x) = 2 \) are \( x = 1 \) and \( x = 2 \). Therefore, we have:
\[ f^{-1}(2) = \{1, 2\} \]