Question

what are the key features of the graph of the absolute value function ?

how do the domain and range of g(x) = a|x| compare to the domain and range of f(x) = |x| when 0 < a < 1 ? Explain ?

Answer 1

Key features of the graph y = |x|:
- V-shaped with vertex at (0,0).
- Symmetric about the y-axis (even function).
- Piecewise linear: |x| = x for x ≥ 0, and |x| = −x for x < 0.
- Slopes: +1 for x > 0, −1 for x < 0 (not differentiable at x = 0).
- Domain: all real numbers (−∞, ∞).
- Range: y ≥ 0, i.e. [0, ∞).
- Continuous everywhere, minimum value 0 at x = 0, unbounded above.

Now compare f(x) = |x| and g(x) = a|x| for 0 < a < 1:
- Domain: the same, (−∞, ∞). Multiplying by a nonzero constant does not restrict x.
- Range: also the same, [0, ∞). Since |x| takes every nonnegative value, g(x) = a|x| takes every nonnegative value as well (for any y ≥ 0 choose x = y/a ≥ 0 so g(x) = y).

Geometric effect when 0 < a < 1:
- Vertical compression toward the x-axis by factor a.
- Slopes become ±a (less steep), vertex remains at (0,0), symmetry and continuity properties remain, and the function is still not differentiable at 0.