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].
### 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].
