Tables of values for two different functions are given below.

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & 4 \\
\hline
2 & 16 \\
\hline
3 & 36 \\
\hline
4 & 64 \\
\hline
5 & 100 \\
\hline
6 & 144 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & 4 \\
\hline
2 & 16 \\
\hline
3 & 64 \\
\hline
4 & 256 \\
\hline
5 & 1,024 \\
\hline
6 & 4,096 \\
\hline
\end{tabular}

Which statement is true?

A. The right function grows approximately 21 times faster than the left function over the interval [tex]$4\ \textless \ x\ \textless \ 5$[/tex].

B. The right function grows approximately 21 times slower than the left function over the interval [tex]$4\ \textless \ x\ \textless \ 5$[/tex].

C. The right function grows approximately 6 times faster than the left function over the interval [tex]$2\ \textless \ x\ \textless \ 3$[/tex].

D. The right function grows approximately 2.5 times slower than the left function over the interval [tex]$2\ \textless \ x\ \textless \ 3$[/tex].



Answer :

Let's analyze the growth rates of the two functions over the specified intervals.

### Interval [tex]\( 4 < x < 5 \)[/tex]

1. For the left function:
At [tex]\( x = 4 \)[/tex], [tex]\( y = 64 \)[/tex]
At [tex]\( x = 5 \)[/tex], [tex]\( y = 100 \)[/tex]

Growth rate for the left function over the interval [tex]\( 4 < x < 5 \)[/tex] is:
[tex]\[ \frac{100}{64} = 1.5625 \][/tex]

2. For the right function:
At [tex]\( x = 4 \)[/tex], [tex]\( y = 256 \)[/tex]
At [tex]\( x = 5 \)[/tex], [tex]\( y = 1024 \)[/tex]

Growth rate for the right function over the interval [tex]\( 4 < x < 5 \)[/tex] is:
[tex]\[ \frac{1024}{256} = 4.0 \][/tex]

3. To find how many times faster the right function grows compared to the left function:
[tex]\[ \frac{4.0}{1.5625} \approx 2.56 \][/tex]

Therefore, the right function grows approximately 2.56 times faster than the left function over the interval [tex]\( 4 < x < 5 \)[/tex].

### Interval [tex]\( 2 < x < 3 \)[/tex]

1. For the left function:
At [tex]\( x = 2 \)[/tex], [tex]\( y = 16 \)[/tex]
At [tex]\( x = 3 \)[/tex], [tex]\( y = 36 \)[/tex]

Growth rate for the left function over the interval [tex]\( 2 < x < 3 \)[/tex] is:
[tex]\[ \frac{36}{16} = 2.25 \][/tex]

2. For the right function:
At [tex]\( x = 2 \)[/tex], [tex]\( y = 16 \)[/tex]
At [tex]\( x = 3 \)[/tex], [tex]\( y = 64 \)[/tex]

Growth rate for the right function over the interval [tex]\( 2 < x < 3 \)[/tex] is:
[tex]\[ \frac{64}{16} = 4.0 \][/tex]

3. To find how many times faster the right function grows compared to the left function:
[tex]\[ \frac{4.0}{2.25} \approx 1.78 \][/tex]

Therefore, the right function grows approximately 1.78 times faster than the left function over the interval [tex]\( 2 < x < 3 \)[/tex].

Based on the analysis, the correct statement is:
- The right function grows approximately 2.56 times faster than the left function over the interval [tex]\( 4 < x < 5 \)[/tex].