Let's break this down step by step.

a. Determine a rule for the function $f(t)$.

We're told that Julia drives at a constant speed. So, the function $f(t)$ gives the distance Julia is from Denver after $t$ hours of driving. We can model this situation using a linear equation, because her speed is constant.

We know the following two pieces of information:

• After 0 hours, she is at Denver, so $f(0) = 0$.
• After 0.5 hours (30 minutes), she is 155 miles away from Denver, so $f(0.5) = 155$.
• After 12 hours, she is 260 miles away, so $f(12) = 260$.

We can find the constant speed (rate of change) by calculating the slope of the line connecting these two points: $(0.5, 155)$ and $(12, 260)$.

The formula for slope is:

$\text{slope} = \frac{\text{change in distance}}{\text{change in time}} = \frac{260 - 155}{12 - 0.5} = \frac{105}{11.5} \approx 9.13$

Thus, Julia's speed is approximately $9.13$ miles per hour. Therefore, the linear function that describes the distance from Denver as a function of time is:

$f(t) = 9.13t$

b. Interpret $f^{-1}(500)$.

The expression $f^{-1}(500)$ refers to the time it takes for Julia to be 500 miles from Denver. To find this, we need to solve for $t$ when $f(t) = 500$:

$9.13t = 500$

Solve for $t$:

$t = \frac{500}{9.13} \approx 54.8 \text{ hours}$

So, $f^{-1}(500) \approx 54.8$ hours. This means it will take about 54.8 hours for Julia to be 500 miles away from Denver.

c. Determine a rule for $f^{-1}$.

To find the inverse function $f^{-1}$, we start with the equation for $f(t)$:

$f(t) = 9.13t$

To find $f^{-1}$, we need to solve for $t$ in terms of $d$ (distance from Denver):

$d = 9.13t$

Solving for $t$:

$t = \frac{d}{9.13}$

So the inverse function is:

$f^{-1}(d) = \frac{d}{9.13}$

d. Construct a graph for $f^{-1}(d)$.

The graph of $f^{-1}(d)$ is a straight line with a slope of $\frac{1}{9.13}$ (approximately $0.1095$). It represents the time it takes to be $d$ miles from Denver.

The graph will have the following properties:

• The y-axis represents time $t$.
• The x-axis represents distance $d$.
• The slope is positive, indicating that as distance increases, time increases.

Now, I can generate a graph for you. Would you like me to do that? Or do you have any further questions about these steps?