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 find the range of \( f(g(x)) \), we need to understand the range of \( g(x) \) and how \( f(x) \) behaves over that range. Let's break this down step-by-step.  

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 \( x \in [0,1) \), \( g(x) = x \) implies the range is \( [0,1) \).
   - For \( x \in (-3,0) \), \( g(x) = -x \) implies the range is \( (0,3) \).
   Therefore, combining both parts, the range of \( g(x) \) is \((0, 3)\).
3. Find \( f(g(x)) \) using the range of \( g(x) \):
   - Since the range of \( g(x) \) is \((0, 3)\), we evaluate \( f(x) \) for \( x \in (0, 3) \).
   - For \( x \in [0,3) \), \( f(x) = 1 - \frac{x}{3} \).
4. Compute the range for \( 1 - \frac{x}{3} \) where \( x \in (0, 3) \):
   - Calculate the endpoints:
   At \( x = 0 \), \( 1 - \frac{0}{3} = 1 \).
   At \( x \rightarrow 3 \), \( 1 - \frac{3}{3} = 0 \). - Thus, the range of \( 1 - \frac{x}{3} \) for \( x \in (0, 3) \) is \((0, 1]\).
5. Conclusion: The range of \( f(g(x)) \) is \((0, 1]\), which matches option

(0, 1]

1. .

The correct answer is: (0, 1].

Answer 2

To find the range of \( f(g(x)) \), we start by evaluating \( g(x) \) and then substitute it into \( f(x) \) according to the intervals provided.

Evaluate \( g(x) \):
\(g(x) =  \begin{cases}    x, & x \in [0, 1] \\   -x, & x \in (-3, 0)  \end{cases}\)
For \( x \in [0, 1] \), \( g(x) = x \) which gives \( g(x) \in [0, 1] \).
For \( x \in (-3, 0) \), \( g(x) = -x \) which gives \( g(x) \in (0, 3] \).
Therefore, the range of \( g(x) \) is \((0, 3]\).

Since \( g(x) \in (0, 3] \), we use the definition of \( f(x) \) for \( x \in [0, 3] \):
\(f(g(x)) = 1 - \frac{g(x)}{3}\)

Determine the range of \( f(g(x)) \) by substituting values from the range of \( g(x) \). For \( g(x) = 0 \), \( f(g(x)) = 1 - \frac{0}{3} = 1 \). For \( g(x) = 3 \), \( f(g(x)) = 1 - \frac{3}{3} = 0 \). Thus, as \( g(x) \) varies over the interval \((0, 3]\), \( f(g(x)) \) varies over the interval \([0, 1]\).

The range of \( f(g(x)) \) is \([0, 1]\).