Find the domain and range of `f(x)=-|x|`.

Answer

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

### Step 1: Understand the function
The function \( f(x) = -|x| \) represents the negative of the absolute value of \( x \). The absolute value function \( |x| \) is always non-negative, meaning it outputs values that are either zero or positive. Therefore, when we take the negative of that, \( -|x| \), the output will always be zero or negative.

### Step 2: Determine the domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined.

- The absolute value function \( |x| \) is defined for all real numbers.
- Since \( f(x) = -|x| \) is also defined for all real numbers, the domain of \( f(x) \) is all real numbers.

**Domain:**
\[ \text{Domain of } f(x) = (-\infty, \infty) \]

### Step 3: Determine the range
The range of a function is the set of all possible output values (y-values) that the function can produce.

- The output of \( |x| \) is always non-negative (i.e., \( |x| \geq 0 \)).
- Therefore, \( -|x| \) will always be less than or equal to zero (i.e., \( -|x| \leq 0 \)).
- The maximum value of \( f(x) \) occurs when \( x = 0 \), where \( f(0) = -|0| = 0 \).
- As \( |x| \) increases (as \( x \) moves away from zero), \( -|x| \) decreases without bound.

**Range:**
\[ \text{Range of } f(x) = (-\infty, 0] \]

### Final Answer
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( (-\infty, 0] \)

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