Let `f(x) = x^(2) - 3x + 2` be a function defined from `R rarr R` and `|x| = {{:(x,, x ge 0),(-x ,, x le 0):}`, then answer the following
Number of solutions of the equation `|f|x|| = 2`

Options

A. 2
B. 3
C. 4
D. 8

Answer

To find the number of solutions for the equation \( |f(|x|)| = 2 \), where \( f(x) = x^2 - 3x + 2 \), we will follow these steps:

### Step 1: Analyze the function \( f(x) \)

The function \( f(x) = x^2 - 3x + 2 \) is a quadratic function. We can factor it to find its roots:

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

This tells us that the roots of the function are \( x = 1 \) and \( x = 2 \).

### Step 2: Determine the vertex of the parabola

The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). Here, \( a = 1 \) and \( b = -3 \):

\[ x = -\frac{-3}{2 \cdot 1} = \frac{3}{2} \]

Now, substituting \( x = \frac{3}{2} \) back into \( f(x) \):

\[ f\left(\frac{3}{2}\right) = \left(\frac{3}{2}\right)^2 - 3\left(\frac{3}{2}\right) + 2 = \frac{9}{4} - \frac{9}{2} + 2 = \frac{9}{4} - \frac{18}{4} + \frac{8}{4} = -\frac{1}{4} \]

### Step 3: Analyze \( |f(x)| \)

Since \( f(x) \) is a quadratic function that opens upwards, it will be below the x-axis between its roots (1 and 2). Therefore, the graph of \( |f(x)| \) will reflect the part of the graph below the x-axis above the x-axis.

### Step 4: Set up the equation \( |f(|x|)| = 2 \)

We need to find the values of \( x \) such that \( |f(|x|)| = 2 \). This leads to two cases:

1. \( f(|x|) = 2 \)
2. \( f(|x|) = -2 \)

### Step 5: Solve \( f(|x|) = 2 \)

Setting \( f(x) = 2 \):

\[ x^2 - 3x + 2 = 2 \implies x^2 - 3x = 0 \implies x(x - 3) = 0 \]

The solutions are \( x = 0 \) and \( x = 3 \).

### Step 6: Solve \( f(|x|) = -2 \)

Setting \( f(x) = -2 \):

\[ x^2 - 3x + 2 = -2 \implies x^2 - 3x + 4 = 0 \]

Calculating the discriminant:

\[ D = b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7 \]

Since the discriminant is negative, there are no real solutions for this case.

### Step 7: Consider the absolute value

Since \( |x| \) can be either positive or negative, the solutions \( x = 0 \) and \( x = 3 \) imply:

- From \( |x| = 0 \), we have \( x = 0 \).
- From \( |x| = 3 \), we have \( x = 3 \) and \( x = -3 \).

### Conclusion

Thus, the total number of solutions to the equation \( |f(|x|)| = 2 \) is:

\[ \text{Total solutions} = 3 \quad (x = 0, x = 3, x = -3) \]

### Final Answer

The number of solutions is **3**.

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