Certainly! Let's work through each part of the question step-by-step.
### a. Complete the Table
We are given the parking rate of $5 per hour. We need to calculate the cost (G) for the given times in hours (t).
[tex]\[
\begin{array}{|c|c|}
\hline
t \text{ (hours)} & G \text{ (dollars)} \\
\hline
0 & 0 \\
\hline
\frac{1}{2} & 2.5 \\
\hline
1 & 5 \\
\hline
1 \frac{3}{4} & 8.75 \\
\hline
2 & 10 \\
\hline
5 & 25 \\
\hline
\end{array}
\][/tex]
These values are calculated as follows:
- For [tex]\( t = 0 \)[/tex], [tex]\( G = 5 \times 0 = 0 \)[/tex].
- For [tex]\( t = \frac{1}{2} \)[/tex], [tex]\( G = 5 \times 0.5 = 2.5 \)[/tex].
- For [tex]\( t = 1 \)[/tex], [tex]\( G = 5 \times 1 = 5 \)[/tex].
- For [tex]\( t = 1 \frac{3}{4} \)[/tex], [tex]\( G = 5 \times 1.75 = 8.75 \)[/tex].
- For [tex]\( t = 2 \)[/tex], [tex]\( G = 5 \times 2 = 10 \)[/tex].
- For [tex]\( t = 5 \)[/tex], [tex]\( G = 5 \times 5 = 25 \)[/tex].
### b. Sketch a Graph of [tex]\( G \)[/tex]
To sketch the graph of [tex]\( G \)[/tex] for [tex]\( 0 \leq t \leq 12 \)[/tex]:
1. Plot the points: [tex]\((0, 0), (0.5, 2.5), (1, 5), (1.75, 8.75), (2, 10), (5, 25)\)[/tex], and include a few more points, such as [tex]\( (10, 50) \)[/tex] and [tex]\( (12, 60) \)[/tex].
2. Draw a straight line through these points because [tex]\( G(t) \)[/tex] is a linear function of [tex]\( t \)[/tex] with a slope of 5.
### c. Is [tex]\( G \)[/tex] a Function of [tex]\( t \)[/tex]?
Yes, [tex]\( G \)[/tex] is a function of [tex]\( t \)[/tex]. In a function, each input value [tex]\( t \)[/tex] must correspond to exactly one output value [tex]\( G(t) \)[/tex]. Here, for every specific number of hours [tex]\( t \)[/tex], there is a unique cost [tex]\( G \)[/tex] calculated as [tex]\( G = 5t \)[/tex]. Therefore, [tex]\( G \)[/tex] satisfies the definition of a function of [tex]\( t \)[/tex].
### d. Is [tex]\( t \)[/tex] a Function of [tex]\( G \)[/tex]?
No, [tex]\( t \)[/tex] is not a function of [tex]\( G \)[/tex]. For [tex]\( t \)[/tex] to be a function of [tex]\( G \)[/tex], each dollar amount [tex]\( G \)[/tex] would need to map to exactly one value of [tex]\( t \)[/tex]. However, because [tex]\( G \)[/tex] can have the same value for different [tex]\( t \)[/tex] values (for example, if another payment plan allowed rounding or discounts), [tex]\( t \)[/tex] does not uniquely correspond to each [tex]\( G \)[/tex].
### Is this a Graph of a Function? Explain Your Reasoning.
Yes, the graph of [tex]\( G \)[/tex] vs [tex]\( t \)[/tex] is a graph of a function. To be a function, the graph must pass the vertical line test – meaning that no vertical line should intersect the graph at more than one point. In this case, since [tex]\( G(t) = 5t \)[/tex] is a linear function, any vertical line will intersect the line at exactly one point, confirming that it is indeed the graph of a function.
