Question

If \(f(x) =   \begin{cases}     2+2x  & ;x∈(-1,0)\\     1-\frac x3  & ;x∈[0,3)  \end{cases}\) and \(g(x) =   \begin{cases}     x  & ;x∈[0,1)\\     -x & ;x∈(-3,0)  \end{cases}\). Then range of \(fog(x)\) is

• A. [0, 1]
• B. [-1,1]
• C. (0,1]
• D. (-1,1)

Answer 1

The Correct Option is C

To determine the range of the composite function \( f(g(x)) \), we first establish the range of the inner function \( g(x) \) and then analyze the behavior of the outer function \( f(x) \) over that determined range. This process is detailed below.

1. Define the functions \( f(x) \) and \( g(x) \):
   \(f(x) = \begin{cases} 2 + 2x & , x \in (-1,0) \\ 1 - \frac{x}{3} & , x \in [0,3) \end{cases}\)
   \(g(x) = \begin{cases} x & , x \in [0,1) \\ -x & , x \in (-3,0) \end{cases}\)
2. Determine the range of \( g(x) \):
   - For the interval \( x \in [0,1) \), \( g(x) = x \), resulting in a range of \( [0,1) \).
   - For the interval \( x \in (-3,0) \), \( g(x) = -x \), resulting in a range of \( (0,3) \).
   The combined range of \( g(x) \) across both intervals is \((0, 3)\).
3. Compute \( f(g(x)) \) using the range of \( g(x) \):
   - Given that the range of \( g(x) \) is \((0, 3)\), we need to evaluate \( f(x) \) for \( x \in (0, 3) \).
   - Within this interval, the relevant definition of \( f(x) \) is \( f(x) = 1 - \frac{x}{3} \) for \( x \in [0,3) \).
4. Calculate the range of \( 1 - \frac{x}{3} \) for \( x \in (0, 3) \):
   - Evaluate the function at the interval boundaries:
   When \( x = 0 \), \( 1 - \frac{0}{3} = 1 \).
   As \( x \) approaches \( 3 \), \( 1 - \frac{3}{3} = 0 \).
   Therefore, for \( x \in (0, 3) \), the range of \( 1 - \frac{x}{3} \) is \((0, 1]\).
5. Conclusion: The range of \( f(g(x)) \) is \((0, 1]\). This corresponds to option

(0, 1]

1. .

The correct answer is: (0, 1].