Asked by Sigma
What are the domain and range of this exponential function?
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In general for a real exponential function y = A·b^(x−h) + k with b>0, b≠1:
- Domain: all real numbers, (−∞, ∞).
- Range: depends on A and k:
- If A > 0, range is (k, ∞).
- If A < 0, range is (−∞, k).
- If A = 0, the function is constant and range is {k}.
Reason: the exponential part b^(x−h) is always positive, so A·b^(x−h) takes values either >0 (if A>0) or <0 (if A<0), and adding k shifts the horizontal asymptote to y = k.
Examples:
- y = 3·2^(x−1) + 4 → domain (−∞, ∞), range (4, ∞).
- y = −2·(1/3)^x − 5 → domain (−∞, ∞), range (−∞, −5).
Share the exact function or graph and I’ll give the specific domain and range.
In general for a real exponential function y = A·b^(x−h) + k with b>0, b≠1:
- Domain: all real numbers, (−∞, ∞).
- Range: depends on A and k:
- If A > 0, range is (k, ∞).
- If A < 0, range is (−∞, k).
- If A = 0, the function is constant and range is {k}.
Reason: the exponential part b^(x−h) is always positive, so A·b^(x−h) takes values either >0 (if A>0) or <0 (if A<0), and adding k shifts the horizontal asymptote to y = k.
Examples:
- y = 3·2^(x−1) + 4 → domain (−∞, ∞), range (4, ∞).
- y = −2·(1/3)^x − 5 → domain (−∞, ∞), range (−∞, −5).
Share the exact function or graph and I’ll give the specific domain and range.
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