Let `f(x)` be a function whose domain is `[-5,6]`. Let `g(x)=|2x+5|`,then domain of `(fog)(x)` is

Options

A. `[-4,1]`
B. `[-5,1]`
C. `[-11/2,1/2]`
D. `[-5,7]`

Explanation

To find the domain of the composite function \( (f \circ g)(x) \), where \( g(x) = |2x + 5| \) and the domain of \( f(x) \) is \([-5, 6]\), we will follow these steps:

### Step 1: Determine the range of \( g(x) \)

The function \( g(x) = |2x + 5| \) will always yield non-negative values because it is an absolute value function. Therefore, the range of \( g(x) \) is:

\[ g(x) \geq 0 \]

### Step 2: Set the range of \( g(x) \) within the domain of \( f(x) \)

Since \( f(x) \) is defined for \( x \) in the interval \([-5, 6]\), we need to find the values of \( x \) such that \( g(x) \) falls within this interval. This means we need to solve the inequalities:

\[ -5 \leq g(x) \leq 6 \]

Since \( g(x) \) is always non-negative, we only need to consider the upper bound:

\[ g(x) \leq 6 \]

### Step 3: Solve the inequality \( |2x + 5| \leq 6 \)

To solve this inequality, we will break it down into two cases based on the definition of absolute value:

1. \( 2x + 5 \leq 6 \)
2. \( 2x + 5 \geq -6 \)

#### Case 1: \( 2x + 5 \leq 6 \)

Subtracting 5 from both sides gives:

\[ 2x \leq 1 \]

Dividing by 2:

\[ x \leq \frac{1}{2} \]

#### Case 2: \( 2x + 5 \geq -6 \)

Subtracting 5 from both sides gives:

\[ 2x \geq -11 \]

Dividing by 2:

\[ x \geq -\frac{11}{2} \]

### Step 4: Combine the results

From the two cases, we have:

\[ -\frac{11}{2} \leq x \leq \frac{1}{2} \]

### Conclusion

Thus, the domain of the composite function \( (f \circ g)(x) \) is:

\[ \left[-\frac{11}{2}, \frac{1}{2}\right] \]