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