Let `f(x) = x^(2) - 5x + 6, g(x) = f(|x|), h(x) = |g(x)|`
The set of values of `x` such that equation `g(x) + |g(x)| = 0` is satisfied contains

Options

A. `-3`
B. `0`
C. `2`
D. `(5)/(2)`

Explanation

To solve the problem, we need to find the set of values of \( x \) such that the equation \( g(x) + |g(x)| = 0 \) is satisfied. Let's break this down step by step.

### Step 1: Define the function \( f(x) \)
We start with the function:
\[ f(x) = x^2 - 5x + 6 \]
We can factor this quadratic equation:
\[ f(x) = (x - 3)(x - 2) \]
This shows that the roots of \( f(x) \) are \( x = 2 \) and \( x = 3 \).

### Step 2: Define the function \( g(x) \)
Next, we define \( g(x) \):
\[ g(x) = f(|x|) = (|x| - 3)(|x| - 2) \]
This means we need to evaluate \( g(x) \) based on the absolute value of \( x \).

### Step 3: Determine the expression for \( g(x) \)
We can analyze \( g(x) \) for different cases based on the value of \( x \):
- For \( x \geq 0 \):
\[ g(x) = (x - 3)(x - 2) \]
- For \( x < 0 \):
\[ g(x) = (-x - 3)(-x - 2) = (3 + x)(2 + x) \]

### Step 4: Define the function \( h(x) \)
Now, we define \( h(x) \):
\[ h(x) = |g(x)| \]
This means \( h(x) \) will take the absolute value of \( g(x) \).

### Step 5: Set up the equation \( g(x) + |g(x)| = 0 \)
The equation we need to solve is:
\[ g(x) + |g(x)| = 0 \]
This can be rewritten as:
\[ g(x) + h(x) = 0 \]
Since \( h(x) = |g(x)| \), this implies:
\[ g(x) + g(x) = 0 \quad \text{if } g(x) \geq 0 \]
or
\[ g(x) + (-g(x)) = 0 \quad \text{if } g(x) < 0 \]
The first case gives no solutions, so we focus on the second case:
\[ g(x) = 0 \]

### Step 6: Solve \( g(x) = 0 \)
We need to find the values of \( x \) such that:
\[ g(x) = 0 \]
From our earlier definitions:
- For \( x \geq 0 \):
\[ (x - 3)(x - 2) = 0 \quad \Rightarrow \quad x = 2, 3 \]
- For \( x < 0 \):
\[ (3 + x)(2 + x) = 0 \quad \Rightarrow \quad x = -3, -2 \]

### Step 7: Combine the results
The solutions to \( g(x) = 0 \) are:
\[ x = -3, -2, 2, 3 \]

### Final Answer
The set of values of \( x \) such that the equation \( g(x) + |g(x)| = 0 \) is satisfied is:
\[ \{ -3, -2, 2, 3 \} \]