Answer :
To answer the given question, we need to analyze the mathematical properties of the functions [tex]\( f(x) \)[/tex], [tex]\( g(x) \)[/tex], and [tex]\( h(x) \)[/tex], as well as compare their behaviors as [tex]\( x \)[/tex] approaches infinity and over the interval [0,2].
### Functions Definition
Given:
1. [tex]\( f(x) = 8x^2 + 3 \)[/tex]
2. [tex]\( g(x) = 9x^2 - 4 \)[/tex]
3. [tex]\( h(x) = 3x + 6 \)[/tex]
### Behavior as [tex]\( x \)[/tex] Approaches Infinity
First, we need to determine the values of these functions as [tex]\( x \)[/tex] approaches infinity.
- For [tex]\( f(x) = 8x^2 + 3 \)[/tex], as [tex]\( x \)[/tex] becomes very large, the term [tex]\( 8x^2 \)[/tex] will dominate, and the function value will be very large. Specifically, [tex]\( f(\infty) \approx 8 \cdot \infty^2 \)[/tex].
- Result: Approximately [tex]\( 8,000,000,000,003 \)[/tex].
- For [tex]\( g(x) = 9x^2 - 4 \)[/tex], as [tex]\( x \)[/tex] becomes very large, the term [tex]\( 9x^2 \)[/tex] will dominate, and the function value will be very large. Specifically, [tex]\( g(\infty) \approx 9 \cdot \infty^2 \)[/tex].
- Result: Approximately [tex]\( 8,999,999,999,996 \)[/tex].
- For [tex]\( h(x) = 3x + 6 \)[/tex], as [tex]\( x \)[/tex] becomes very large, the term [tex]\( 3x \)[/tex] will dominate. However, this will grow much slower than the quadratic terms in [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
- Result: Approximately [tex]\( 3,000,006 \)[/tex].
Comparing these values:
- [tex]\( g(x) \)[/tex] is significantly larger than both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex].
- [tex]\( f(x) \)[/tex] is much larger than [tex]\( h(x) \)[/tex].
Thus, as [tex]\( x \)[/tex] approaches infinity, [tex]\( g(x) \)[/tex] will exceed both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex], and [tex]\( f(x) \)[/tex] will exceed [tex]\( h(x) \)[/tex].
### Average Rate of Change Over the Interval [0, 2]
Next, we calculate the average rate of change for each function over the interval [0,2].
- For [tex]\( f(x) \)[/tex]:
[tex]\[ \text{Average Rate of Change}_f = \frac{f(2) - f(0)}{2 - 0} = \frac{(8 \cdot 2^2 + 3) - (8 \cdot 0^2 + 3)}{2} = \frac{35 - 3}{2} = 16 \][/tex]
- For [tex]\( g(x) \)[/tex]:
[tex]\[ \text{Average Rate of Change}_g = \frac{g(2) - g(0)}{2 - 0} = \frac{(9 \cdot 2^2 - 4) - (9 \cdot 0^2 - 4)}{2} = \frac{32 - (-4)}{2} = 18 \][/tex]
- For [tex]\( h(x) \)[/tex]:
[tex]\[ \text{Average Rate of Change}_h = \frac{h(2) - h(0)}{2 - 0} = \frac{(3 \cdot 2 + 6) - (3 \cdot 0 + 6)}{2} = \frac{12 - 6}{2} = 3 \][/tex]
Comparing these rates of change:
- The average rate of change for [tex]\( f \)[/tex] is 16.
- The average rate of change for [tex]\( g \)[/tex] is 18.
- The average rate of change for [tex]\( h \)[/tex] is 3.
Thus, the average rate of change of both [tex]\( f \)[/tex] and [tex]\( h \)[/tex] is less than the average rate of change of [tex]\( g \)[/tex].
### Conclusion
From the above analysis, we can determine the true statements:
- Correct: As [tex]\( x \)[/tex] approaches infinity, the value of [tex]\( g(x) \)[/tex] eventually exceeds the values of both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex].
- Correct: Over the interval [0, 2], the average rate of change of [tex]\( f \)[/tex] and [tex]\( h \)[/tex] is less than the average rate of change of [tex]\( g \)[/tex].
- Incorrect: As [tex]\( x \)[/tex] approaches infinity, the values of [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex] eventually exceed the value of [tex]\( f(x) \)[/tex].
Thus, the following statements are true:
- [tex]\( C \)[/tex]: As [tex]\( x \)[/tex] approaches infinity, the value of [tex]\( g(x) \)[/tex] eventually exceeds the values of both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex].
- [tex]\( B \)[/tex]: Over the interval [0, 2], the average rate of change of [tex]\( f \)[/tex] and [tex]\( h \)[/tex] is less than the average rate of change of [tex]\( g \)[/tex].
### Functions Definition
Given:
1. [tex]\( f(x) = 8x^2 + 3 \)[/tex]
2. [tex]\( g(x) = 9x^2 - 4 \)[/tex]
3. [tex]\( h(x) = 3x + 6 \)[/tex]
### Behavior as [tex]\( x \)[/tex] Approaches Infinity
First, we need to determine the values of these functions as [tex]\( x \)[/tex] approaches infinity.
- For [tex]\( f(x) = 8x^2 + 3 \)[/tex], as [tex]\( x \)[/tex] becomes very large, the term [tex]\( 8x^2 \)[/tex] will dominate, and the function value will be very large. Specifically, [tex]\( f(\infty) \approx 8 \cdot \infty^2 \)[/tex].
- Result: Approximately [tex]\( 8,000,000,000,003 \)[/tex].
- For [tex]\( g(x) = 9x^2 - 4 \)[/tex], as [tex]\( x \)[/tex] becomes very large, the term [tex]\( 9x^2 \)[/tex] will dominate, and the function value will be very large. Specifically, [tex]\( g(\infty) \approx 9 \cdot \infty^2 \)[/tex].
- Result: Approximately [tex]\( 8,999,999,999,996 \)[/tex].
- For [tex]\( h(x) = 3x + 6 \)[/tex], as [tex]\( x \)[/tex] becomes very large, the term [tex]\( 3x \)[/tex] will dominate. However, this will grow much slower than the quadratic terms in [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
- Result: Approximately [tex]\( 3,000,006 \)[/tex].
Comparing these values:
- [tex]\( g(x) \)[/tex] is significantly larger than both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex].
- [tex]\( f(x) \)[/tex] is much larger than [tex]\( h(x) \)[/tex].
Thus, as [tex]\( x \)[/tex] approaches infinity, [tex]\( g(x) \)[/tex] will exceed both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex], and [tex]\( f(x) \)[/tex] will exceed [tex]\( h(x) \)[/tex].
### Average Rate of Change Over the Interval [0, 2]
Next, we calculate the average rate of change for each function over the interval [0,2].
- For [tex]\( f(x) \)[/tex]:
[tex]\[ \text{Average Rate of Change}_f = \frac{f(2) - f(0)}{2 - 0} = \frac{(8 \cdot 2^2 + 3) - (8 \cdot 0^2 + 3)}{2} = \frac{35 - 3}{2} = 16 \][/tex]
- For [tex]\( g(x) \)[/tex]:
[tex]\[ \text{Average Rate of Change}_g = \frac{g(2) - g(0)}{2 - 0} = \frac{(9 \cdot 2^2 - 4) - (9 \cdot 0^2 - 4)}{2} = \frac{32 - (-4)}{2} = 18 \][/tex]
- For [tex]\( h(x) \)[/tex]:
[tex]\[ \text{Average Rate of Change}_h = \frac{h(2) - h(0)}{2 - 0} = \frac{(3 \cdot 2 + 6) - (3 \cdot 0 + 6)}{2} = \frac{12 - 6}{2} = 3 \][/tex]
Comparing these rates of change:
- The average rate of change for [tex]\( f \)[/tex] is 16.
- The average rate of change for [tex]\( g \)[/tex] is 18.
- The average rate of change for [tex]\( h \)[/tex] is 3.
Thus, the average rate of change of both [tex]\( f \)[/tex] and [tex]\( h \)[/tex] is less than the average rate of change of [tex]\( g \)[/tex].
### Conclusion
From the above analysis, we can determine the true statements:
- Correct: As [tex]\( x \)[/tex] approaches infinity, the value of [tex]\( g(x) \)[/tex] eventually exceeds the values of both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex].
- Correct: Over the interval [0, 2], the average rate of change of [tex]\( f \)[/tex] and [tex]\( h \)[/tex] is less than the average rate of change of [tex]\( g \)[/tex].
- Incorrect: As [tex]\( x \)[/tex] approaches infinity, the values of [tex]\( g(x) \)[/tex] and [tex]\( h(x) \)[/tex] eventually exceed the value of [tex]\( f(x) \)[/tex].
Thus, the following statements are true:
- [tex]\( C \)[/tex]: As [tex]\( x \)[/tex] approaches infinity, the value of [tex]\( g(x) \)[/tex] eventually exceeds the values of both [tex]\( f(x) \)[/tex] and [tex]\( h(x) \)[/tex].
- [tex]\( B \)[/tex]: Over the interval [0, 2], the average rate of change of [tex]\( f \)[/tex] and [tex]\( h \)[/tex] is less than the average rate of change of [tex]\( g \)[/tex].
