A single formula that gives `f(x)` for all `x ge0` where `f(x)={(3+x, 0lexlt3),(3x-3, xge3):}` is

Options

A. `f(x)=|2x-1|+4x`
B. `f(x)=|x-3|+2x`
C. `f(x)=|3x-9|-x`
D. `f(x)=|x-3|+3x`

Answer

To find a single formula that gives \( f(x) \) for all \( x \geq 0 \), we start with the piecewise function defined as:

\[ f(x) = \begin{cases} 3 + x & \text{if } 0 \leq x < 3 \\ 3x - 3 & \text{if } x \geq 3 \end{cases} \]

### Step 1: Analyze the piecewise function
We have two cases to consider based on the value of \( x \):
1. For \( 0 \leq x < 3 \), \( f(x) = 3 + x \)
2. For \( x \geq 3 \), \( f(x) = 3x - 3 \)

### Step 2: Identify the critical point
The critical point where the definition of the function changes is at \( x = 3 \). We need to ensure that our single formula correctly represents both cases at this point.

### Step 3: Check the function values at the critical point
- For \( x = 3 \):
- From the first case: \( f(3) = 3 + 3 = 6 \)
- From the second case: \( f(3) = 3(3) - 3 = 9 - 3 = 6 \)

Both cases yield the same value at \( x = 3 \), which is \( 6 \).

### Step 4: Construct the single formula using absolute value
To combine both cases into a single formula, we can use the absolute value function. The absolute value function can help us express the piecewise nature of \( f(x) \).

The function can be rewritten as:
\[ f(x) = |x - 3| + 3x \]

### Step 5: Verify the combined formula
Now we will check if this formula matches the original piecewise function for both cases.

- **Case 1**: When \( 0 \leq x < 3 \):
- \( |x - 3| = 3 - x \)
- Thus, \( f(x) = (3 - x) + 3x = 3 + 2x \)
- This does not match the original function \( 3 + x \).

- **Case 2**: When \( x \geq 3 \):
- \( |x - 3| = x - 3 \)
- Thus, \( f(x) = (x - 3) + 3x = 4x - 3 \)
- This does not match the original function \( 3x - 3 \).

### Step 6: Find the correct representation
After analyzing the options given, we can check option B:
\[ f(x) = |x - 3| + 2x \]

- **For \( 0 \leq x < 3 \)**:
- \( |x - 3| = 3 - x \)
- Thus, \( f(x) = (3 - x) + 2x = 3 + x \)

- **For \( x \geq 3 \)**:
- \( |x - 3| = x - 3 \)
- Thus, \( f(x) = (x - 3) + 2x = 3x - 3 \)

Both cases match the original piecewise function.

### Final Answer:
The single formula that gives \( f(x) \) for all \( x \geq 0 \) is:
\[ f(x) = |x - 3| + 2x \]