Question

For which type of function does the average rate of change produce the same value for any interval of the domain on the function? Why does this occur for this type of function?(2 points)

Answer 1

The average rate of change produces the same value for any interval of the domain for **linear functions**.

### Explanation:
A linear function can be expressed in the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The average rate of change between two points \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \) is calculated as:

\[
\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}
\]

For a linear function:

\[
f(x_1) = mx_1 + b \quad \text{and} \quad f(x_2) = mx_2 + b
\]

Substituting these into the average rate of change formula gives:

\[
\text{Average Rate of Change} = \frac{(mx_2 + b) - (mx_1 + b)}{x_2 - x_1} = \frac{mx_2 - mx_1}{x_2 - x_1} = \frac{m(x_2 - x_1)}{x_2 - x_1} = m
\]

Since \( m \) (the slope) is constant, the average rate of change remains the same for any interval in the domain of the function. This consistency occurs because linear functions have a constant slope, leading to a uniform change in the function value relative to changes in the input.