Answer :
To determine which pair of functions consists of an exponential function that consistently grows faster than the quadratic function over the interval [tex]\(0 \leq x \leq 5\)[/tex], we analyze the behavior and growth rates of each pair.
We are given four pairs of functions:
1. Pair 1: [tex]\( f_1(x) = 2x^2 + 3x + 5 \)[/tex] and [tex]\( g_1(x) = e^{0.5x} \)[/tex]
2. Pair 2: [tex]\( f_2(x) = x^2 + 2x + 1 \)[/tex] and [tex]\( g_2(x) = e^x \)[/tex]
3. Pair 3: [tex]\( f_3(x) = 3x^2 + x + 1 \)[/tex] and [tex]\( g_3(x) = e^{0.3x} \)[/tex]
4. Pair 4: [tex]\( f_4(x) = 4x^2 + 2x + 7 \)[/tex] and [tex]\( g_4(x) = e^{x/2} \)[/tex]
We need to determine which exponential function [tex]\( g(x) \)[/tex] consistently grows at a faster rate than its corresponding quadratic function [tex]\( f(x) \)[/tex] throughout the entire interval [tex]\(0 \leq x \leq 5\)[/tex].
Let's check each pair:
### Pair 1: [tex]\( f_1(x) = 2x^2 + 3x + 5 \)[/tex] and [tex]\( g_1(x) = e^{0.5x} \)[/tex]
For various values of [tex]\(x\)[/tex] from 0 to 5:
- When [tex]\(x = 0\)[/tex], [tex]\( f_1(0) = 5\)[/tex] and [tex]\( g_1(0) = 1 \)[/tex]
- When [tex]\(x = 1\)[/tex], [tex]\( f_1(1) = 10\)[/tex] and [tex]\( g_1(1) \approx 1.65 \)[/tex]
- When [tex]\(x = 5\)[/tex], [tex]\( f_1(5) = 70\)[/tex] and [tex]\( g_1(5) \approx 12.18 \)[/tex]
Clearly, for Pair 1, [tex]\( f_1(x) > g_1(x) \)[/tex] for larger values of [tex]\(x\)[/tex].
### Pair 2: [tex]\( f_2(x) = x^2 + 2x + 1 \)[/tex] and [tex]\( g_2(x) = e^x \)[/tex]
For various values of [tex]\(x\)[/tex] from 0 to 5:
- When [tex]\(x = 0\)[/tex], [tex]\( f_2(0) = 1 \)[/tex] and [tex]\( g_2(0) = 1 \)[/tex]
- When [tex]\(x = 1\)[/tex], [tex]\( f_2(1) = 4 \)[/tex] and [tex]\( g_2(1) \approx 2.72 \)[/tex]
- When [tex]\(x = 5\)[/tex], [tex]\( f_2(5) = 36\)[/tex] and [tex]\( g_2(5) \approx 148.41 \)[/tex]
Clearly, for Pair 2, [tex]\( g_2(x) > f_2(x) \)[/tex] for larger values of [tex]\(x\)[/tex].
### Pair 3: [tex]\( f_3(x) = 3x^2 + x + 1 \)[/tex] and [tex]\( g_3(x) = e^{0.3x} \)[/tex]
For various values of [tex]\(x\)[/tex] from 0 to 5:
- When [tex]\(x = 0\)[/tex], [tex]\( f_3(0) = 1 \)[/tex] and [tex]\( g_3(0) = 1 \)[/tex]
- When [tex]\(x = 1\)[/tex], [tex]\( f_3(1) = 5 \)[/tex] and [tex]\( g_3(1) \approx 1.35 \)[/tex]
- When [tex]\(x = 5\)[/tex], [tex]\( f_3(5) = 81\)[/tex] and [tex]\( g_3(5) \approx 4.48 \)[/tex]
Clearly, for Pair 3, [tex]\( f_3(x) < g_3(x) \)[/tex] for larger values of [tex]\(x\)[/tex].
### Pair 4: [tex]\( f_4(x) = 4x^2 + 2x + 7 \)[/tex] and [tex]\( g_4(x) = e^{x/2} \)[/tex]
For various values of [tex]\(x\)[/tex] from 0 to 5:
- When [tex]\(x = 0\)[/tex], [tex]\( f_4(0) = 7 \)[/tex] and [tex]\( g_4(0) = 1 \)[/tex]
- When [tex]\(x = 1\)[/tex], [tex]\( f_4(1) = 13 \)[/tex] and [tex]\( g_4(1) \approx 1.65 \)[/tex]
- When [tex]\(x = 5\)[/tex], [tex]\( f_4(5) = 117\)[/tex] and [tex]\( g_4(5) \approx 12.18 \)[/tex]
Clearly, for Pair 4, [tex]\( f_4(x) > g_4(x) \)[/tex] for larger values of [tex]\(x\)[/tex].
From this analysis, none of the pairs exhibit an exponential function that consistently grows faster than the quadratic function over the entire interval [tex]\(0 \leq x \leq 5\)[/tex].
Thus, there is no pair for which the exponential function consistently grows at a faster rate than the quadratic function over the interval [tex]\(0 \leq x \leq 5\)[/tex].
We are given four pairs of functions:
1. Pair 1: [tex]\( f_1(x) = 2x^2 + 3x + 5 \)[/tex] and [tex]\( g_1(x) = e^{0.5x} \)[/tex]
2. Pair 2: [tex]\( f_2(x) = x^2 + 2x + 1 \)[/tex] and [tex]\( g_2(x) = e^x \)[/tex]
3. Pair 3: [tex]\( f_3(x) = 3x^2 + x + 1 \)[/tex] and [tex]\( g_3(x) = e^{0.3x} \)[/tex]
4. Pair 4: [tex]\( f_4(x) = 4x^2 + 2x + 7 \)[/tex] and [tex]\( g_4(x) = e^{x/2} \)[/tex]
We need to determine which exponential function [tex]\( g(x) \)[/tex] consistently grows at a faster rate than its corresponding quadratic function [tex]\( f(x) \)[/tex] throughout the entire interval [tex]\(0 \leq x \leq 5\)[/tex].
Let's check each pair:
### Pair 1: [tex]\( f_1(x) = 2x^2 + 3x + 5 \)[/tex] and [tex]\( g_1(x) = e^{0.5x} \)[/tex]
For various values of [tex]\(x\)[/tex] from 0 to 5:
- When [tex]\(x = 0\)[/tex], [tex]\( f_1(0) = 5\)[/tex] and [tex]\( g_1(0) = 1 \)[/tex]
- When [tex]\(x = 1\)[/tex], [tex]\( f_1(1) = 10\)[/tex] and [tex]\( g_1(1) \approx 1.65 \)[/tex]
- When [tex]\(x = 5\)[/tex], [tex]\( f_1(5) = 70\)[/tex] and [tex]\( g_1(5) \approx 12.18 \)[/tex]
Clearly, for Pair 1, [tex]\( f_1(x) > g_1(x) \)[/tex] for larger values of [tex]\(x\)[/tex].
### Pair 2: [tex]\( f_2(x) = x^2 + 2x + 1 \)[/tex] and [tex]\( g_2(x) = e^x \)[/tex]
For various values of [tex]\(x\)[/tex] from 0 to 5:
- When [tex]\(x = 0\)[/tex], [tex]\( f_2(0) = 1 \)[/tex] and [tex]\( g_2(0) = 1 \)[/tex]
- When [tex]\(x = 1\)[/tex], [tex]\( f_2(1) = 4 \)[/tex] and [tex]\( g_2(1) \approx 2.72 \)[/tex]
- When [tex]\(x = 5\)[/tex], [tex]\( f_2(5) = 36\)[/tex] and [tex]\( g_2(5) \approx 148.41 \)[/tex]
Clearly, for Pair 2, [tex]\( g_2(x) > f_2(x) \)[/tex] for larger values of [tex]\(x\)[/tex].
### Pair 3: [tex]\( f_3(x) = 3x^2 + x + 1 \)[/tex] and [tex]\( g_3(x) = e^{0.3x} \)[/tex]
For various values of [tex]\(x\)[/tex] from 0 to 5:
- When [tex]\(x = 0\)[/tex], [tex]\( f_3(0) = 1 \)[/tex] and [tex]\( g_3(0) = 1 \)[/tex]
- When [tex]\(x = 1\)[/tex], [tex]\( f_3(1) = 5 \)[/tex] and [tex]\( g_3(1) \approx 1.35 \)[/tex]
- When [tex]\(x = 5\)[/tex], [tex]\( f_3(5) = 81\)[/tex] and [tex]\( g_3(5) \approx 4.48 \)[/tex]
Clearly, for Pair 3, [tex]\( f_3(x) < g_3(x) \)[/tex] for larger values of [tex]\(x\)[/tex].
### Pair 4: [tex]\( f_4(x) = 4x^2 + 2x + 7 \)[/tex] and [tex]\( g_4(x) = e^{x/2} \)[/tex]
For various values of [tex]\(x\)[/tex] from 0 to 5:
- When [tex]\(x = 0\)[/tex], [tex]\( f_4(0) = 7 \)[/tex] and [tex]\( g_4(0) = 1 \)[/tex]
- When [tex]\(x = 1\)[/tex], [tex]\( f_4(1) = 13 \)[/tex] and [tex]\( g_4(1) \approx 1.65 \)[/tex]
- When [tex]\(x = 5\)[/tex], [tex]\( f_4(5) = 117\)[/tex] and [tex]\( g_4(5) \approx 12.18 \)[/tex]
Clearly, for Pair 4, [tex]\( f_4(x) > g_4(x) \)[/tex] for larger values of [tex]\(x\)[/tex].
From this analysis, none of the pairs exhibit an exponential function that consistently grows faster than the quadratic function over the entire interval [tex]\(0 \leq x \leq 5\)[/tex].
Thus, there is no pair for which the exponential function consistently grows at a faster rate than the quadratic function over the interval [tex]\(0 \leq x \leq 5\)[/tex].
