If a funciton f(x) is defined for ` x in [0,1]`, then the function f(2x+3) is defined for

Options

A. `x in [0,1]`
B. `x in [-3//2,-1]`
C. `x in R`
D. `x in [-3//2,1]`

Answer

To determine the interval for which the function \( f(2x + 3) \) is defined given that \( f(x) \) is defined for \( x \in [0, 1] \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the range of \( f(x) \)**:
We know that \( f(x) \) is defined for \( x \) in the interval \([0, 1]\). This means that for \( f(2x + 3) \) to be defined, the expression \( 2x + 3 \) must also fall within the interval \([0, 1]\).

2. **Set up the inequality**:
We need to find the values of \( x \) such that:
\[ 0 \leq 2x + 3 \leq 1 \]

3. **Break it into two inequalities**:
We can break this compound inequality into two separate inequalities:
- \( 2x + 3 \geq 0 \)
- \( 2x + 3 \leq 1 \)

4. **Solve the first inequality**:
For the first inequality \( 2x + 3 \geq 0 \):
\[ 2x \geq -3 \]
\[ x \geq -\frac{3}{2} \]

5. **Solve the second inequality**:
For the second inequality \( 2x + 3 \leq 1 \):
\[ 2x \leq 1 - 3 \]
\[ 2x \leq -2 \]
\[ x \leq -1 \]

6. **Combine the results**:
From the two inequalities, we have:
\[ -\frac{3}{2} \leq x \leq -1 \]

7. **Write the interval**:
In interval notation, this can be expressed as:
\[ x \in \left[-\frac{3}{2}, -1\right] \]

### Final Answer:
The function \( f(2x + 3) \) is defined for \( x \in \left[-\frac{3}{2}, -1\right] \).

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