find the range of the function f, of, x, equals, vertical bar, x, minus, 5, vertical bar, .f(x)=∣x−5∣.

The function given is \( f(x) = |x - 5| \).

### Step-by-Step Analysis:

1. **Understanding the Function:**
   - The function represents the absolute value of \( x - 5 \).
   - Absolute value functions output only non-negative values.

2. **Range of the Function:**
   - Since \( f(x) = |x - 5| \) represents an absolute value, the smallest value \( f(x) \) can take is \( 0 \), which occurs when \( x = 5 \).
   - For any other value of \( x \), \( |x - 5| \) will always be positive.

   Therefore, \( f(x) \geq 0 \) for all \( x \).

3. **Conclusion:**
   - The range of the function \( f(x) = |x - 5| \) is \( [0, \infty) \).

Would you like further details on absolute value functions or have any additional questions?

### Relative Questions:
1. What is the domain of the function \( f(x) = |x - 5| \)?
2. How would shifting the function \( f(x) = |x - 5| \) affect its range?
3. How does the graph of \( f(x) = |x - 5| \) compare to the graph of \( f(x) = |x| \)?
4. Can an absolute value function ever have a negative output?
5. How would you find the minimum value of a more complex absolute value function?

**Tip:** Remember that the vertex of the graph of an absolute value function, \( f(x) = |x - h| \), occurs at \( x = h \).

Discover how to find the range of the absolute value function |x - 5|. Understand the behavior of absolute value functions and explore why the range of f(x) = |x - 5| is [0, ∞). Suitable for high school students studying absolute value functions.

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