Question

Algebra 1, 9th grade


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. Compare g(x) = a|x| with f(x) = |x| when 0 < a < 1
- Domain: the same — all real numbers, (-∞, ∞).
- Range: both are [0, ∞). Multiplying by 0 < a < 1 does not change the lower bound 0 or the fact that values grow without bound as |x| → ∞.
- What does change: g is a vertical shrink of f. For each x ≠ 0, 0 < g(x) < f(x) — the graph is closer to the x-axis than y = |x|.

4. Janiece’s error about the vertex of g(x) = a|x|
- If a > 0 the V opens upward and the vertex (0,0) is the minimum.
- If a < 0 the V opens downward and the vertex is the maximum.
- If a = 0 the function is constant g(x)=0, so the vertex is both min and max.
So the vertex is a minimum only when a > 0.

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

5. g(x) = 5|x|
- Domain: (-∞, ∞)
- Range: [0, ∞)
- Sign: g(0) = 0. For x ≠ 0, g(x) > 0. So positive on (-∞,0) ∪ (0,∞); never negative.
- Piecewise form: g(x) = {5x if x ≥ 0; -5x if x < 0}.
- Graph: V-shaped with vertex at (0,0). Right branch is line y = 5x (slope 5); left branch is y = -5x (slope -5). Example points: g(1)=5, g(-1)=5.

6. h(x) = -2|x|
- Domain: (-∞, ∞)
- Range: (-∞, 0]
- Sign: h(0) = 0. For x ≠ 0, h(x) < 0. So negative on (-∞,0) ∪ (0,∞); never positive.
- Piecewise form: h(x) = {-2x if x ≥ 0; 2x if x < 0}.
- Graph: inverted V with vertex at (0,0). Right branch is line y = -2x (slope -2); left branch is y = 2x (slope 2). Example points: h(1) = -2, h(-1) = -2.

7. g(x) = 1.5|x|
- Domain: (-∞, ∞)
- Range: [0, ∞)
- Sign: zero at x = 0, positive for x ≠ 0.
- Piecewise: g(x) = {1.5x if x ≥ 0; -1.5x if x < 0}.
- Graph: V with vertex (0,0). Right branch y = 1.5x, left branch y = -1.5x. Example: g(1)=1.5, g(-1)=1.5.

8. h(x) = -0.8|x|
- Domain: (-∞, ∞)
- Range: (-∞, 0]
- Sign: zero at x = 0, negative for x ≠ 0.
- Piecewise: h(x) = {-0.8x if x ≥ 0; 0.8x if x < 0}.
- Graph: inverted V with vertex (0,0). Right branch y = -0.8x, left branch y = 0.8x. Example: h(1) = -0.8, h(-1) = -0.8.

If you want, I can sketch these four graphs or give a table of sample points to plot.