The range of the function æ’(x) = |x| is all non‑negative real numbers, because absolute value never returns a negative result.
Understanding this range helps clarify how input values map to outputs and why the graph sits above the x‑axis.
| Input x | Expression æ’(x) | Output y | Position on graph |
|---|---|---|---|
| -3 | |-3| | 3 | Point in second quadrant |
| -1 | |-1| | 1 | Point above negative x |
| 0 | |0| | 0 | Vertex at origin |
| 2 | |2| | 2 | Point in first quadrant |
| 5 | |5| | 5 | Point further right |
Behavior for positive inputs
When x is positive or zero, the expression æ’(x) = |x| simplifies to x itself.
This segment of the function is a straight line with slope one, starting at the origin and extending into the first quadrant.
Each increase in x produces an identical increase in output, so the positive side directly reflects the input values.
Behavior for negative inputs
For negative x, the absolute value reverses the sign, turning any negative number into its positive counterpart.
Graphically, this flips the left side of the graph to mirror the right side, creating a symmetric V shape.
No matter how negative the input, the output remains zero or positive, reinforcing the non‑negative nature of the range.
Vertex and minimum value
The lowest point of the graph occurs at x = 0, where æ’(x) = |x| equals zero.
This vertex anchors the V shape and confirms that zero is included in the range.
Because the arms of the graph rise indefinitely, there is no upper bound on the output values.
Graphical representation and visual confirmation
Plotting y = |x| shows a sharp V with the vertex at the origin and arms extending upward in both directions.
The graph never dips below the x‑axis, which visually confirms that the range is limited to zero and positive numbers.
Each vertical line intersects the graph at most once, illustrating that the function is well defined for every real input.
Key takeaways for the range of absolute value functions
- The range of æ’(x) = |x| is all real numbers y such that y ≥ 0.
- Absolute value functions produce non‑negative outputs regardless of the sign of the input.
- The vertex of the graph corresponds to the minimum value in the range.
- Symmetry about the y‑axis confirms that both positive and negative inputs map to the same set of outputs.
- Understanding this range supports accurate graphing, solving inequalities, and modeling real situations with distance.
FAQ
Reader questions
Can the output of æ’(x) = |x| ever be negative?
No, the absolute value ensures the output is zero or positive for every real input.
Is zero included in the range of æ’(x) = |x|?
Yes, when x is zero, the function outputs zero, so zero belongs to the range.
Does the range include fractions and decimals?
Yes, any non‑negative rational or irrational number can appear as an output, such as 0.5 or √2.
Are there any restrictions on the size of the output values?
No, the outputs can grow without bound, so the range extends to positive infinity.