Question

a remote - control car moves toward a wall, stops, backs up, and stops. the graph shows the car’s distance from the wall as a function of time. answers parts a through d.
a. write the domain and range in inequalities. identify the domain. select the correct choice below and fill in the answer box(es) to complete your choice.
a. ( leq y leq )
b. ( x leq )
c. ( leq x )
d. ( y leq )
e. ( leq x leq )
f. ( leq y )

Explanation:

To solve the problem of finding the domain (time) for the remote - control car's motion, we analyze the x - axis (time) of the graph:

Step 1: Understand the domain concept

The domain of a function (in this case, the distance from the wall as a function of time) represents the set of all possible input values (time values, \(x\)). From the graph, we can see the time starts at \(x = 0\) (when the motion begins) and ends at some maximum value of \(x\) (when the car stops its final motion).

Step 2: Determine the domain inequality form

Looking at the options, the domain is related to the time \(x\). The correct form for the domain (time) inequality should be in terms of \(x\). The general form for the domain of a function with a minimum value \(a\) and maximum value \(b\) is \(a\leq x\leq b\). From the graph (even though the exact numbers are a bit unclear from the image, the structure of the problem and the options suggest that the domain is in the form of \( \leq x\leq\) (option E - \( \leq x\leq\)) or we can infer that the time starts at 0 and goes up to a certain value. But since the options are about the form, and the domain is for \(x\) (time), the correct option for the domain inequality form is E ( \( \leq x\leq\)) (assuming the graph has a start and end time for \(x\)).

For the range (distance \(y\) from the wall)

The range is the set of all possible output values (distance \(y\)). The car moves toward the wall (decreasing \(y\)) and then backs up (increasing \(y\)), so the range will have a minimum and maximum value. But for the inequality form, if we consider the range in terms of \(y\), the correct form would be similar to \( \leq y\leq\) (but the options for range are also given). However, the question first asks for the domain's inequality form.

If we assume the time (domain) starts at 0 and ends at, say, 10 seconds (just as an example to fit the form), the domain inequality would be \(0\leq x\leq10\) (in the form of \( \leq x\leq\), which is option E).

Answer:

E. \( \leq x\leq\) (We would fill in the specific numbers from the graph, but based on the form of the options, E is the correct form for the domain inequality as it is in the form \(a\leq x\leq b\) which is used for the domain of a function with a minimum and maximum \(x\) - value)