The domain and range of the functions given by `f(x)=2-|x-5|` are

Options

A. Domain `=R^(+)`, Range `=(-oo,1]`
B. Domain =R Range `=(-oo,2]`
C. Domain `=R^(+)`, Range `=(-oo,2]`
D. Domain =R Range `=(-oo,2]`

Answer

To find the domain and range of the function \( f(x) = 2 - |x - 5| \), we will follow these steps:

### Step 1: Determine the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined.

For the function \( f(x) = 2 - |x - 5| \):
- The expression \( |x - 5| \) is defined for all real numbers \( x \).
- Therefore, \( f(x) \) is also defined for all real numbers.

**Domain:**
\[ \text{Domain of } f(x) = \mathbb{R} \quad \text{(all real numbers)} \]

### Step 2: Determine the Range
The range of a function is the set of all possible output values (y-values) that the function can produce.

1. **Analyze the absolute value**: The expression \( |x - 5| \) is always non-negative, meaning:
\[ |x - 5| \geq 0 \]

2. **Substituting into the function**: Since \( |x - 5| \geq 0 \), we can write:
\[ -|x - 5| \leq 0 \]

3. **Adding 2 to both sides**:
\[ 2 - |x - 5| \leq 2 \]
This shows that the maximum value of \( f(x) \) is 2.

4. **Finding the minimum value**: The minimum value of \( f(x) \) occurs when \( |x - 5| \) is maximized. As \( |x - 5| \) can increase indefinitely as \( x \) moves away from 5, \( f(x) \) can decrease without bound:
\[ \text{As } |x - 5| \to \infty, \quad f(x) \to -\infty \]

5. **Conclusion about the range**: Therefore, the range of \( f(x) \) is all values less than or equal to 2:
\[ \text{Range: } (-\infty, 2] \]

### Final Answer:
- **Domain**: \( \mathbb{R} \) (all real numbers)
- **Range**: \( (-\infty, 2] \)

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