The type of function for which the average rate of change produces the same value for any interval of the domain is a Linear function.
Why does this occur for linear functions?
A linear function can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y-intercept. The average rate of change of a linear function over any interval \([x_1, x_2]\) can be calculated using the formula:
\[ \text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
Substituting the linear function into this formula gives:
\[ \text{Average Rate of Change} = \frac{(mx_2 + b) - (mx_1 + b)}{x_2 - x_1} = \frac{mx_2 - mx_1}{x_2 - x_1} = m \]
In this case, the average rate of change simplifies to just \( m \), the slope of the linear function, which is constant regardless of the choice of \( x_1 \) and \( x_2 \). Consequently, the average rate of change is the same for any interval on the linear function's domain.
In contrast:
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Quadratic functions (e.g., \( f(x) = ax^2 + bx + c \)) have a variable rate of change that depends on the specific interval, as the slope increases or decreases depending on the values of \( x \).
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Exponential functions (e.g., \( f(x) = a \cdot b^x \)) also have a varying rate of change that changes based on the base and the specific values of \( x \) over intervals.
Therefore, only linear functions produce a constant average rate of change across any interval in their domain.