Question

• no more than 9 miles per hour over: $151
• more than 9 miles per hour to no more than 14 miles
• more than 14 miles per hour to no more than 19 miles per hour over
• more than 19 miles per hour to no more than 29 miles per hour over: $301
• more than 29 miles per hour over: $376

write a step function that represents the relationship between the number ( x ) of miles per hour over the speed limit and the
f(x) = \begin{cases} 151, & \text{if } 0 < x leq 9 \\ 226, & \text{if } 9 < x leq 14 \\ 276, & \text{if } square \\ 301, & \text{if } 19 < x leq 29 \\ square, & \text{if } square end{cases}

Explanation:

Step1: Analyze the speed ranges

We have the following speed over limit ranges and corresponding fines:

• \(0 < x \leq 9\): \(\$151\)
• \(9 < x \leq 14\): \(\$226\)
• Next, we need to find the range for the fine of \(\$276\). From the given information, the next range after \(9 < x \leq 14\) should be \(14 < x \leq 19\) (since the subsequent range is \(19 < x \leq 29\) with fine \(\$301\)).
• Then \(19 < x \leq 29\): \(\$301\)
• And for more than 29 miles per hour over, the fine is \(\$376\), so the range is \(x > 29\).

Step2: Determine the range for \(\$276\)

Looking at the sequence of speed ranges, after \(9 < x \leq 14\) (fine \(\$226\)) and before \(19 < x \leq 29\) (fine \(\$301\)), the range should be \(14 < x \leq 19\) for the fine of \(\$276\).

Step3: Determine the range for \(\$376\)

For the fine of \(\$376\), the range is \(x > 29\) (or \(29 < x\)).

Answer:

The step function \(f(x)\) is:

\[
f(x) =

$$\begin{cases} 151, & \text{if } 0 < x \leq 9 \\ 226, & \text{if } 9 < x \leq 14 \\ 276, & \text{if } 14 < x \leq 19 \\ 301, & \text{if } 19 < x \leq 29 \\ 376, & \text{if } x > 29 \end{cases}$$

\]

So the missing range for \(276\) is \(14 < x \leq 19\) and for \(376\) is \(x > 29\). If we are only filling the box for the \(276\) case, the answer is \(14 < x \leq 19\).