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Select the correct answer.

Jordan bought an antique clock. The net value of the clock is equal to the resale value of the clock minus Jordan's cost to buy it. The exponential function [tex]f[/tex], shown in the table, represents the net value of the antique clock, rounded to the nearest whole dollar, [tex]x[/tex] years after Jordan purchased it.

\begin{tabular}{|l|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & 0 & 1 & 2 & 3 & 4 \\
\hline
[tex]$f(x)$[/tex] & -50 & -26 & 0 & 29 & 61 \\
\hline
\end{tabular}

Which statement is true?

A. During the first two years Jordan owned the clock, the resale value decreased. Then it increased after year 2.

B. 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.

C. During the first two years Jordan owned the clock, the resale value increased. Then it decreased after year 2.

D. During the first two years Jordan owned the clock, the resale value exceeded the cost. Then it was less than the cost after year 2.



Answer :

GinnyAnswer
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]