If rangr of funtion `f (x)` whose domain is set of all real numbers is `[-2,4],` then range of function `g (x)=1/2 f (2x+1)` is equal to

Options

A. `[-2, 4]`
B. `[-1, 2]`
C. `[-3, 9]`
D. `[-2, 2]`

Answer

To find the range of the function \( g(x) = \frac{1}{2} f(2x + 1) \) given that the range of \( f(x) \) is \([-2, 4]\), we can follow these steps:

### Step 1: Understand the transformation of the function
The function \( g(x) \) is derived from \( f(x) \) by two transformations:
1. The input \( x \) is transformed by \( 2x + 1 \).
2. The output of \( f(2x + 1) \) is then scaled by \( \frac{1}{2} \).

### Step 2: Determine the range of \( f(2x + 1) \)
Since the domain of \( f(x) \) is all real numbers, the transformation \( 2x + 1 \) will also cover all real numbers as \( x \) varies over all reals. Therefore, the range of \( f(2x + 1) \) will remain the same as the range of \( f(x) \), which is \([-2, 4]\).

### Step 3: Scale the range by \( \frac{1}{2} \)
Now we need to scale the range of \( f(2x + 1) \) by \( \frac{1}{2} \):
- The minimum value of the range of \( f(2x + 1) \) is \(-2\).
- The maximum value of the range of \( f(2x + 1) \) is \(4\).

Now we apply the scaling:
- Minimum: \( \frac{1}{2} \times (-2) = -1 \)
- Maximum: \( \frac{1}{2} \times 4 = 2 \)

### Step 4: Write the final range of \( g(x) \)
Thus, the range of \( g(x) \) is \([-1, 2]\).

### Final Answer:
The range of the function \( g(x) = \frac{1}{2} f(2x + 1) \) is \([-1, 2]\).

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