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

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]`

Explanation

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.

1. The function \( f(x) = 2 - |x - 5| \) involves the absolute value function \( |x - 5| \).
2. The absolute value function is defined for all real numbers. Therefore, there are no restrictions on the values of \( x \).
3. Thus, the domain of \( f(x) \) is all real numbers.

**Domain:**
\[ \text{Domain} = \{ x \in \mathbb{R} \} \]

### Step 2: Determine the Range
The range of a function is the set of all possible output values (f(x)-values).

1. The expression \( |x - 5| \) represents the distance between \( x \) and 5, which is always non-negative. Therefore, the minimum value of \( |x - 5| \) is 0, occurring when \( x = 5 \).
2. As \( x \) moves away from 5, \( |x - 5| \) increases without bound. Therefore, the maximum value of \( |x - 5| \) approaches infinity.
3. The function can be rewritten to analyze its maximum and minimum values:
\[ f(x) = 2 - |x - 5| \]
4. The maximum value of \( f(x) \) occurs when \( |x - 5| \) is at its minimum (which is 0):
\[ f(5) = 2 - 0 = 2 \]
5. The minimum value of \( f(x) \) occurs when \( |x - 5| \) is at its maximum (which approaches infinity):
\[ \lim_{|x - 5| \to \infty} f(x) = 2 - \infty = -\infty \]
6. Therefore, the range of \( f(x) \) is all values from negative infinity up to 2.

**Range:**
\[ \text{Range} = (-\infty, 2] \]

### Final Answer
- **Domain:** \( \{ x \in \mathbb{R} \} \)
- **Range:** \( (-\infty, 2] \)

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