The function f(x) = 4x has constant percentage rate of change on all intervals of length one.

The statement regarding the function $f(x) = 4x$ indicates that this function has a constant percentage rate of change, which can be analyzed further.

Understanding the Concept

1. Definition of Constant Percentage Rate of Change: A function has a constant percentage rate of change if the ratio of the change in the function's value to the original value remains the same as the input changes.

2. Identifying the Rate of Change: For a linear function like $f(x) = 4x$, the rate of change is linear and can be expressed as: $\text{Rate of change} = \frac{f(x + 1) - f(x)}{1} = \frac{4(x + 1) - 4x}{1} = 4$

Analysis

1. Calculating the Percentage Change: For small changes $\Delta x = 1$, we can find the percentage change: $\text{Percentage change} = \frac{4}{f(x)} \times 100\%$ Substituting $f(x)$: $\text{Percentage change} = \frac{4}{4x} \times 100\% = \frac{100}{x} \%$

2. Interpretation: The percentage change depends on $x$. Therefore, while the nominal rate of change (in terms of value) is constant at 4, the percentage change is not constant across different values of $x$. Thus, the function does not have a constant percentage rate of change.

Final Summary

While $f(x) = 4x$ maintains a constant nominal rate of change, its percentage rate of change varies with $x$, confirming that it does not have a constant percentage rate of change across all intervals of length one.