Question

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

4. Janiece says that the vertex of the graph of g(x) = a|x| always represents the minimum value of the function g. Explain her error.

For each function, find the domain, the range, and the intervals over which it is positive or negative.

5. g(x) = 5|x|

6. h(x) = -2|x|

Graph each function.

7. g(x) = 1.5|x|

8. h(x) = -0.8|x|

Answer 1

2. Comparison when 0 < a < 1 for g(x) = a|x| vs f(x) = |x|
- Domain: both are all real numbers, (−∞, ∞).
- Range: both are [0, ∞). (Since a>0, a|x| ≥ 0 for all x and can grow without bound as |x| → ∞.)
- Effect on the graph: multiplying by 0<a<1 vertically compresses the graph toward the x-axis (it is less steep than |x|), but the vertex (0,0) and the domain and range remain the same.

4. Janiece’s error about the vertex always being a minimum for g(x)=a|x|
- If a>0 the vertex (0,0) is a minimum (smallest value).
- If a<0 the vertex (0,0) is a maximum (largest value), because the V is flipped downward.
- If a = 0 the function is constant 0 (everywhere equally minimum and maximum).
So you must consider the sign of a; the vertex is not always a minimum.

For each function give domain, range, and where positive/negative

5. g(x) = 5|x|
- Domain: (−∞, ∞)
- Range: [0, ∞)
- Positive: g(x) > 0 for x ≠ 0, i.e. on (−∞, 0) ∪ (0, ∞)
- Zero at x = 0
- Never negative
- Shape: V with slopes −5 for x>0 and +5 for x<0, vertex at (0,0). Example points: (0,0), (1,5), (−1,5).

6. h(x) = −2|x|
- Domain: (−∞, ∞)
- Range: (−∞, 0]
- Positive: never (h(x) > 0 on ∅)
- Zero at x = 0
- Negative: h(x) < 0 for x ≠ 0, i.e. on (−∞, 0) ∪ (0, ∞)
- Shape: inverted V with vertex (0,0) (a maximum). For x≥0 slope is −2, for x<0 slope is +2. Example points: (0,0), (1,−2), (−1,−2).

Graphs (key points and shape)

7. g(x) = 1.5|x|
- Domain: all real numbers; Range: [0, ∞)
- Vertex: (0,0). Slopes: +1.5 for x<0, −1.5 for x>0.
- Example points to plot: (0,0), (1,1.5), (2,3), (−1,1.5), (−2,3).
- V-shape, vertically stretched relative to |x| by factor 1.5 (steeper).

8. h(x) = −0.8|x|
- Domain: all real numbers; Range: (−∞, 0]
- Vertex: (0,0). Slopes: +0.8 for x<0, −0.8 for x>0.
- Example points to plot: (0,0), (1,−0.8), (2,−1.6), (−1,−0.8), (−2,−1.6).
- Inverted V, less steep than |x| (vertical compression), maximum at (0,0).