Let `f(x)= sin^(-1)x and g(x) = (x^2-x-2)/(2x^2-x-6)`. If `g(2)=underset(xrarr2)(lim)g(x)` then the domain of the function fog is

Options

A. `(-oo,-2]uu[-3/2,oo)`
B. `(-oo,-2]uu[-1,oo)`
C. `(-oo,-2]uu[-4/3,oo)`
D. `(-oo,-1]uu[2,oo)`

Answer

To find the domain of the function \( f(g(x)) \), we need to follow these steps:

### Step 1: Define the Functions
We have:
- \( f(x) = \sin^{-1}(x) \)
- \( g(x) = \frac{x^2 - x - 2}{2x^2 - x - 6} \)

### Step 2: Find \( g(2) \) and \( \lim_{x \to 2} g(x) \)
First, we need to evaluate \( g(2) \):
\[ g(2) = \frac{2^2 - 2 - 2}{2(2^2) - 2 - 6} = \frac{4 - 2 - 2}{8 - 2 - 6} = \frac{0}{0} \]
Since this results in an indeterminate form, we will find \( \lim_{x \to 2} g(x) \).

### Step 3: Factor \( g(x) \)
We can factor the numerator and denominator of \( g(x) \):
- The numerator \( x^2 - x - 2 \) factors to \( (x - 2)(x + 1) \).
- The denominator \( 2x^2 - x - 6 \) factors to \( (2x + 3)(x - 2) \).

Thus, we can rewrite \( g(x) \):
\[ g(x) = \frac{(x - 2)(x + 1)}{(2x + 3)(x - 2)} \]
For \( x \neq 2 \), we can simplify this to:
\[ g(x) = \frac{x + 1}{2x + 3} \]

### Step 4: Find \( \lim_{x \to 2} g(x) \)
Now, we can compute the limit:
\[ \lim_{x \to 2} g(x) = \lim_{x \to 2} \frac{x + 1}{2x + 3} = \frac{2 + 1}{2(2) + 3} = \frac{3}{7} \]
Thus, \( g(2) = \frac{3}{7} \).

### Step 5: Determine the Domain of \( f(g(x)) \)
The function \( f(g(x)) \) is defined when \( g(x) \) is in the range of \( f \), which is \( [-1, 1] \):
\[ -1 \leq g(x) \leq 1 \]
Substituting \( g(x) \):
\[ -1 \leq \frac{x + 1}{2x + 3} \leq 1 \]

### Step 6: Solve the Inequalities
1. **For the left inequality**:
\[ -1 \leq \frac{x + 1}{2x + 3} \]
Cross-multiplying (considering \( 2x + 3 > 0 \) for \( x > -\frac{3}{2} \)):
\[ - (2x + 3) \leq x + 1 \implies -2x - 3 \leq x + 1 \implies -3 - 1 \leq 3x \implies -4 \leq 3x \implies x \geq -\frac{4}{3} \]

2. **For the right inequality**:
\[ \frac{x + 1}{2x + 3} \leq 1 \]
Cross-multiplying:
\[ x + 1 \leq 2x + 3 \implies 1 - 3 \leq 2x - x \implies -2 \leq x \implies x \geq -2 \]

### Step 7: Combine the Results
The combined inequalities are:
\[ x \geq -\frac{4}{3} \quad \text{and} \quad x \geq -2 \]
Thus, the domain of \( f(g(x)) \) is:
\[ x \geq -\frac{4}{3} \]

### Final Domain
The domain of the function \( f(g(x)) \) is:
\[ \boxed{[-\frac{4}{3}, \infty)} \]