Answer :
Let's carefully analyze the given information step by step, focusing on the table of values and interpreting their implications:
[tex]\[ \begin{tabular}{|l|c|c|c|c|c|} \hline $x$ & 0 & 1 & 2 & 3 & 4 \\ \hline $f(x)$ & -50 & -26 & 0 & 29 & 61 \\ \hline \end{tabular} \][/tex]
We are asked to determine which statement is true based on the function [tex]\( f \)[/tex], which represents the net value of the antique clock [tex]\( x \)[/tex] years after it was purchased.
### Step-by-Step Analysis:
1. First Two Years (Year 0 and Year 1):
- [tex]\( f(0) = -50 \)[/tex]
- [tex]\( f(1) = -26 \)[/tex]
- [tex]\( f(2) = 0 \)[/tex]
Let's interpret these values:
- At Year 0, the net value is [tex]\(-50\)[/tex], implying that the resale value of the clock is [tex]$50 less than the cost. - At Year 1, the net value is \(-26\), meaning it is now $[/tex]26 less than the cost.
- At Year 2, the net value reaches [tex]\(0\)[/tex], indicating that the resale value of the clock is equal to the cost.
Observations:
- The net value increased from [tex]\(-50\)[/tex] to [tex]\(-26\)[/tex] during the first year.
- The net value increased again from [tex]\(-26\)[/tex] to [tex]\(0\)[/tex] during the second year.
- During these first two years, the net value increased and was less than the cost.
2. After Year 2 (Year 3 and Year 4):
- [tex]\( f(3) = 29 \)[/tex]
- [tex]\( f(4) = 61 \)[/tex]
Let's interpret these values:
- At Year 3, the net value is [tex]\(29\)[/tex], which means the resale value exceeds the cost by [tex]$29. - At Year 4, the net value is \(61\), indicating that the resale value exceeds the cost by $[/tex]61.
Observation:
- After the second year (Year 2), the net value increased from [tex]\(0\)[/tex] to [tex]\(29\)[/tex] and then to [tex]\(61\)[/tex].
### Conclusions:
- First Two Years:
- The net value was less than the cost ([tex]\(f(0) < 0\)[/tex] and [tex]\(f(1) < 0\)[/tex]).
- The net value increased ([tex]\(f(0) < f(1) < f(2)\)[/tex]).
- After Second Year:
- The net value exceeded the cost, as it transitioned from 0 (in [tex]\(f(2)\)[/tex]) to positive values ([tex]\(29\)[/tex] in [tex]\(f(3)\)[/tex] and [tex]\(61\)[/tex] in [tex]\(f(4)\)[/tex]).
Based on this detailed analysis, we can confirm that statement 2 is true:
"During the first two years Jordan owned the clock, the resale value was less than the cost. Then it exceeded the cost after year 2."
Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
[tex]\[ \begin{tabular}{|l|c|c|c|c|c|} \hline $x$ & 0 & 1 & 2 & 3 & 4 \\ \hline $f(x)$ & -50 & -26 & 0 & 29 & 61 \\ \hline \end{tabular} \][/tex]
We are asked to determine which statement is true based on the function [tex]\( f \)[/tex], which represents the net value of the antique clock [tex]\( x \)[/tex] years after it was purchased.
### Step-by-Step Analysis:
1. First Two Years (Year 0 and Year 1):
- [tex]\( f(0) = -50 \)[/tex]
- [tex]\( f(1) = -26 \)[/tex]
- [tex]\( f(2) = 0 \)[/tex]
Let's interpret these values:
- At Year 0, the net value is [tex]\(-50\)[/tex], implying that the resale value of the clock is [tex]$50 less than the cost. - At Year 1, the net value is \(-26\), meaning it is now $[/tex]26 less than the cost.
- At Year 2, the net value reaches [tex]\(0\)[/tex], indicating that the resale value of the clock is equal to the cost.
Observations:
- The net value increased from [tex]\(-50\)[/tex] to [tex]\(-26\)[/tex] during the first year.
- The net value increased again from [tex]\(-26\)[/tex] to [tex]\(0\)[/tex] during the second year.
- During these first two years, the net value increased and was less than the cost.
2. After Year 2 (Year 3 and Year 4):
- [tex]\( f(3) = 29 \)[/tex]
- [tex]\( f(4) = 61 \)[/tex]
Let's interpret these values:
- At Year 3, the net value is [tex]\(29\)[/tex], which means the resale value exceeds the cost by [tex]$29. - At Year 4, the net value is \(61\), indicating that the resale value exceeds the cost by $[/tex]61.
Observation:
- After the second year (Year 2), the net value increased from [tex]\(0\)[/tex] to [tex]\(29\)[/tex] and then to [tex]\(61\)[/tex].
### Conclusions:
- First Two Years:
- The net value was less than the cost ([tex]\(f(0) < 0\)[/tex] and [tex]\(f(1) < 0\)[/tex]).
- The net value increased ([tex]\(f(0) < f(1) < f(2)\)[/tex]).
- After Second Year:
- The net value exceeded the cost, as it transitioned from 0 (in [tex]\(f(2)\)[/tex]) to positive values ([tex]\(29\)[/tex] in [tex]\(f(3)\)[/tex] and [tex]\(61\)[/tex] in [tex]\(f(4)\)[/tex]).
Based on this detailed analysis, we can confirm that statement 2 is true:
"During the first two years Jordan owned the clock, the resale value was less than the cost. Then it exceeded the cost after year 2."
Thus, the correct answer is:
[tex]\[ \boxed{2} \][/tex]
