Let f be a function whose domain is the set of all real number. If `f(x)=|x|-x`, what is the range of f?

Answer

To find the range of the function \( f(x) = |x| - x \), we will analyze the function based on the properties of the absolute value function.

### Step 1: Understand the absolute value function
The absolute value function \( |x| \) behaves differently depending on whether \( x \) is positive or negative:
- If \( x \geq 0 \), then \( |x| = x \).
- If \( x < 0 \), then \( |x| = -x \).

### Step 2: Define the function for different cases
We will break down the function \( f(x) \) into two cases based on the value of \( x \).

**Case 1:** When \( x \geq 0 \)
\[ f(x) = |x| - x = x - x = 0 \]

**Case 2:** When \( x < 0 \)
\[ f(x) = |x| - x = -x - x = -2x \]
Since \( x \) is negative in this case, \( -2x \) will be positive. As \( x \) approaches 0 from the left, \( -2x \) approaches 0, and as \( x \) goes to negative infinity, \( -2x \) goes to positive infinity.

### Step 3: Determine the range from both cases
From **Case 1**, we found that when \( x \geq 0 \), \( f(x) = 0 \).

From **Case 2**, we found that when \( x < 0 \), \( f(x) = -2x \) which can take any positive value as \( x \) approaches negative infinity. Therefore, \( f(x) \) can take values from \( 0 \) to \( +\infty \).

### Step 4: Combine the results to find the range
Combining both cases, we find that:
- For \( x \geq 0 \), \( f(x) = 0 \).
- For \( x < 0 \), \( f(x) \) can take any positive value.

Thus, the range of \( f(x) \) is:
\[ \text{Range of } f = [0, +\infty) \]

### Conclusion
The range of the function \( f(x) = |x| - x \) is \( [0, +\infty) \).

---