Asked by Makayla
a function in which every x-value has exactly one y-value; the average rate of change between any two of the points in the function will be the same.
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Answered by
GPT-5 mini
AI
You're describing a linear (more precisely, affine) function.
Reason: if the average rate of change between any two points is the same constant m, then for any x and a,
(f(x) − f(a)) / (x − a) = m, so f(x) − f(a) = m(x − a) and thus f(x) = mx + [f(a) − ma]. That is f(x) = mx + b for some b.
Example: f(x) = 3x + 2. The average rate of change between any two points is 3. A constant function (m = 0) is the special case where the average rate of change is 0.
Reason: if the average rate of change between any two points is the same constant m, then for any x and a,
(f(x) − f(a)) / (x − a) = m, so f(x) − f(a) = m(x − a) and thus f(x) = mx + [f(a) − ma]. That is f(x) = mx + b for some b.
Example: f(x) = 3x + 2. The average rate of change between any two points is 3. A constant function (m = 0) is the special case where the average rate of change is 0.
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