Let f(x) be a function whose domain is `[-5, 7] and g(x) = |2x + 5|,` then the domain of fog(x) is (A) `[-5,1]` (B) `[-4,0]` (C) `[-6,1]` (D) none of these

Options

A. `[-4,1]`
B. `[-5,1]`
C. `[-6,1]`
D. `[-3,1]`

Explanation

To find the domain of \( f(g(x)) \), we start with the given information:

1. The function \( g(x) = |2x + 5| \).
2. The domain of \( f(x) \) is \( [-5, 7] \).

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

Since \( g(x) = |2x + 5| \), we first need to find the values of \( x \) that will keep \( g(x) \) within the domain of \( f(x) \).

### Step 2: Set up the inequality for \( g(x) \)

For \( f(g(x)) \) to be defined, \( g(x) \) must lie within the domain of \( f(x) \). Therefore, we need:

\[ -5 \leq g(x) \leq 7 \]

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

\[ g(x) \leq 7 \]

### Step 3: Solve the inequality

Substituting \( g(x) \):

\[ |2x + 5| \leq 7 \]

This absolute value inequality can be split into two cases:

1. \( 2x + 5 \leq 7 \)
2. \( 2x + 5 \geq -7 \)

#### Case 1: \( 2x + 5 \leq 7 \)

Subtracting 5 from both sides:

\[ 2x \leq 2 \]

Dividing by 2:

\[ x \leq 1 \]

#### Case 2: \( 2x + 5 \geq -7 \)

Subtracting 5 from both sides:

\[ 2x \geq -12 \]

Dividing by 2:

\[ x \geq -6 \]

### Step 4: Combine the results

From the two cases, we have:

\[ -6 \leq x \leq 1 \]

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

\[ [-6, 1] \]

### Conclusion

The domain of \( f(g(x)) \) is \( [-6, 1] \), which corresponds to option (C).

---