Find the range of the function `f(x)=|x|`.

Explanation

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

### Step 1: Understanding the Absolute Value Function
The absolute value function \( f(x) = |x| \) is defined as:
- \( f(x) = x \) when \( x \geq 0 \)
- \( f(x) = -x \) when \( x < 0 \)

### Step 2: Evaluating the Function for Non-Negative Values
When \( x \) is greater than or equal to zero:
- For \( x = 0 \), \( f(0) = |0| = 0 \)
- For \( x > 0 \), \( f(x) = x \) which is always positive.

Thus, for all non-negative values of \( x \), \( f(x) \) takes values starting from \( 0 \) and goes to \( +\infty \).

### Step 3: Evaluating the Function for Negative Values
When \( x \) is less than zero:
- For \( x < 0 \), \( f(x) = -x \). Since \( x \) is negative, \( -x \) will be positive.
- Therefore, \( f(x) \) will also yield positive values.

### Step 4: Conclusion on the Range
From the above analysis, we can conclude that:
- The function \( f(x) = |x| \) always produces values that are greater than or equal to \( 0 \).
- The minimum value of \( f(x) \) is \( 0 \) (when \( x = 0 \)), and it can take any positive value as \( x \) increases or decreases.

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