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To determine which function is most likely increasing exponentially, we need to analyze the growth rates of both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
First, let's calculate the growth rates for each function as [tex]\( x \)[/tex] increases:
### Calculating the Growth Rates for [tex]\( f(x) \)[/tex]:
1. For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[ \frac{f(2) - f(1)}{2 - 1} = \frac{5 - 2}{1} = 3.0 \][/tex]
2. For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[ \frac{f(3) - f(2)}{3 - 2} = \frac{10 - 5}{1} = 5.0 \][/tex]
3. For [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]:
[tex]\[ \frac{f(4) - f(3)}{4 - 3} = \frac{17 - 10}{1} = 7.0 \][/tex]
4. For [tex]\( x = 4 \)[/tex] to [tex]\( x = 5 \)[/tex]:
[tex]\[ \frac{f(5) - f(4)}{5 - 4} = \frac{26 - 17}{1} = 9.0 \][/tex]
The growth rates for [tex]\( f(x) \)[/tex] are [tex]\( 3.0, 5.0, 7.0, 9.0 \)[/tex].
### Calculating the Growth Rates for [tex]\( g(x) \)[/tex]:
1. For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[ \frac{g(2) - g(1)}{2 - 1} = \frac{4 - 2}{1} = 2.0 \][/tex]
2. For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[ \frac{g(3) - g(2)}{3 - 2} = \frac{8 - 4}{1} = 4.0 \][/tex]
3. For [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]:
[tex]\[ \frac{g(4) - g(3)}{4 - 3} = \frac{16 - 8}{1} = 8.0 \][/tex]
4. For [tex]\( x = 4 \)[/tex] to [tex]\( x = 5 \)[/tex]:
[tex]\[ \frac{g(5) - g(4)}{5 - 4} = \frac{32 - 16}{1} = 16.0 \][/tex]
The growth rates for [tex]\( g(x) \)[/tex] are [tex]\( 2.0, 4.0, 8.0, 16.0 \)[/tex].
### Analyzing the Growth Rates:
- The growth rates for [tex]\( f(x) \)[/tex] (3.0, 5.0, 7.0, 9.0) increase by a constant amount:
[tex]\[ 5.0 - 3.0 = 2.0 \][/tex]
[tex]\[ 7.0 - 5.0 = 2.0 \][/tex]
[tex]\[ 9.0 - 7.0 = 2.0 \][/tex]
This pattern indicates a quadratic function.
- The growth rates for [tex]\( g(x) \)[/tex] (2.0, 4.0, 8.0, 16.0) double each time:
[tex]\[ \frac{4.0}{2.0} = 2.0 \][/tex]
[tex]\[ \frac{8.0}{4.0} = 2.0 \][/tex]
[tex]\[ \frac{16.0}{8.0} = 2.0 \][/tex]
This pattern indicates an exponential function, as the growth rate multiplies by a constant factor.
### Conclusion:
The function [tex]\( g(x) \)[/tex] grows exponentially since its growth rate multiplies by a constant factor. Thus, the function that grows faster is [tex]\( g(x) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{g(x), \text{ because it grows faster than } f(x).} \][/tex]
First, let's calculate the growth rates for each function as [tex]\( x \)[/tex] increases:
### Calculating the Growth Rates for [tex]\( f(x) \)[/tex]:
1. For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[ \frac{f(2) - f(1)}{2 - 1} = \frac{5 - 2}{1} = 3.0 \][/tex]
2. For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[ \frac{f(3) - f(2)}{3 - 2} = \frac{10 - 5}{1} = 5.0 \][/tex]
3. For [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]:
[tex]\[ \frac{f(4) - f(3)}{4 - 3} = \frac{17 - 10}{1} = 7.0 \][/tex]
4. For [tex]\( x = 4 \)[/tex] to [tex]\( x = 5 \)[/tex]:
[tex]\[ \frac{f(5) - f(4)}{5 - 4} = \frac{26 - 17}{1} = 9.0 \][/tex]
The growth rates for [tex]\( f(x) \)[/tex] are [tex]\( 3.0, 5.0, 7.0, 9.0 \)[/tex].
### Calculating the Growth Rates for [tex]\( g(x) \)[/tex]:
1. For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[ \frac{g(2) - g(1)}{2 - 1} = \frac{4 - 2}{1} = 2.0 \][/tex]
2. For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[ \frac{g(3) - g(2)}{3 - 2} = \frac{8 - 4}{1} = 4.0 \][/tex]
3. For [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]:
[tex]\[ \frac{g(4) - g(3)}{4 - 3} = \frac{16 - 8}{1} = 8.0 \][/tex]
4. For [tex]\( x = 4 \)[/tex] to [tex]\( x = 5 \)[/tex]:
[tex]\[ \frac{g(5) - g(4)}{5 - 4} = \frac{32 - 16}{1} = 16.0 \][/tex]
The growth rates for [tex]\( g(x) \)[/tex] are [tex]\( 2.0, 4.0, 8.0, 16.0 \)[/tex].
### Analyzing the Growth Rates:
- The growth rates for [tex]\( f(x) \)[/tex] (3.0, 5.0, 7.0, 9.0) increase by a constant amount:
[tex]\[ 5.0 - 3.0 = 2.0 \][/tex]
[tex]\[ 7.0 - 5.0 = 2.0 \][/tex]
[tex]\[ 9.0 - 7.0 = 2.0 \][/tex]
This pattern indicates a quadratic function.
- The growth rates for [tex]\( g(x) \)[/tex] (2.0, 4.0, 8.0, 16.0) double each time:
[tex]\[ \frac{4.0}{2.0} = 2.0 \][/tex]
[tex]\[ \frac{8.0}{4.0} = 2.0 \][/tex]
[tex]\[ \frac{16.0}{8.0} = 2.0 \][/tex]
This pattern indicates an exponential function, as the growth rate multiplies by a constant factor.
### Conclusion:
The function [tex]\( g(x) \)[/tex] grows exponentially since its growth rate multiplies by a constant factor. Thus, the function that grows faster is [tex]\( g(x) \)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{g(x), \text{ because it grows faster than } f(x).} \][/tex]
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