If `f(g(a))=6, f(x)=(x)/(2)+2, and g(x)=|x^(2)-10|`, which of the following is a possible value of a?

Options

A. `sqrt(2)`
B. `2`
C. `6`
D. `18`

Answer

To solve the problem, we need to find a value of \( a \) such that \( f(g(a)) = 6 \), where \( f(x) = \frac{x}{2} + 2 \) and \( g(x) = |x^2 - 10| \).

### Step-by-Step Solution:

1. **Set up the equation**: We know that \( f(g(a)) = 6 \). Therefore, we need to find \( g(a) \) such that when it is plugged into \( f(x) \), the result is 6.

2. **Find the value of \( x \) that makes \( f(x) = 6 \)**:
\[ f(x) = \frac{x}{2} + 2 \]
Setting this equal to 6:
\[ \frac{x}{2} + 2 = 6 \]
Subtract 2 from both sides:
\[ \frac{x}{2} = 4 \]
Multiply both sides by 2:
\[ x = 8 \]
So, we need \( g(a) = 8 \).

3. **Set up the equation for \( g(a) \)**:
\[ g(a) = |a^2 - 10| = 8 \]
This gives us two cases to solve:
- Case 1: \( a^2 - 10 = 8 \)
- Case 2: \( a^2 - 10 = -8 \)

4. **Solve Case 1**:
\[ a^2 - 10 = 8 \]
Adding 10 to both sides:
\[ a^2 = 18 \]
Taking the square root:
\[ a = \sqrt{18} \quad \text{or} \quad a = -\sqrt{18} \]
Simplifying:
\[ a = 3\sqrt{2} \quad \text{or} \quad a = -3\sqrt{2} \]

5. **Solve Case 2**:
\[ a^2 - 10 = -8 \]
Adding 10 to both sides:
\[ a^2 = 2 \]
Taking the square root:
\[ a = \sqrt{2} \quad \text{or} \quad a = -\sqrt{2} \]

6. **Possible values of \( a \)**:
From both cases, the possible values of \( a \) are:
- \( a = 3\sqrt{2} \)
- \( a = -3\sqrt{2} \)
- \( a = \sqrt{2} \)
- \( a = -\sqrt{2} \)

7. **Select a possible value**: The question asks for a possible value of \( a \). Among the values calculated, \( \sqrt{2} \) is a valid option.

### Conclusion:
A possible value of \( a \) is \( \sqrt{2} \).