Asked by Tesla
Which is the graph of f (x) = 3 (two-thirds) Superscript x? f(x) = 3 (2/3) ^ x
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Answered by
GPT-5 mini
AI
f(x) = 3(2/3)^x is an exponential decay curve. Key features to identify its graph:
- Domain: (−∞, ∞)
- Range: (0, ∞) (y is always positive)
- y‑intercept: f(0) = 3
- No x‑intercepts (never reaches 0)
- Horizontal asymptote: y = 0
- Monotonic: strictly decreasing (because base 2/3 < 1)
- End behavior: as x → ∞, f(x) → 0; as x → −∞, f(x) → ∞
Some points:
- f(0) = 3
- f(1) = 2
- f(2) = 4/3 ≈ 1.333
- f(−1) = 4.5
- f(−2) = 6.75
So choose the graph that crosses the y‑axis at 3, stays above the x‑axis, decreases left→right and approaches y=0 on the right.
- Domain: (−∞, ∞)
- Range: (0, ∞) (y is always positive)
- y‑intercept: f(0) = 3
- No x‑intercepts (never reaches 0)
- Horizontal asymptote: y = 0
- Monotonic: strictly decreasing (because base 2/3 < 1)
- End behavior: as x → ∞, f(x) → 0; as x → −∞, f(x) → ∞
Some points:
- f(0) = 3
- f(1) = 2
- f(2) = 4/3 ≈ 1.333
- f(−1) = 4.5
- f(−2) = 6.75
So choose the graph that crosses the y‑axis at 3, stays above the x‑axis, decreases left→right and approaches y=0 on the right.
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