Let's analyze the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] and verify the provided statements.
### Function Analysis
#### Function 1: [tex]\( f(x) = 3x + 9 \)[/tex]
1. Maxima or Minima:
- [tex]\( f(x) \)[/tex] is a linear function with a positive slope (3).
- Linear functions do not have a maxima or minima because they extend infinitely in both the positive and negative directions without curving.
- Conclusion: [tex]\( f(x) \)[/tex] does not have a maxima (False).
2. X-Intercept:
- To find the x-intercept, set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[
3x + 9 = 0 \implies 3x = -9 \implies x = -3
\][/tex]
- Therefore, [tex]\( f(x) \)[/tex] intersects the x-axis at [tex]\( x = -3 \)[/tex].
- Conclusion: [tex]\( f(x) \)[/tex] has exactly one x-intercept (True).
#### Function 2: [tex]\( g(x) = -x^2 + 9 \)[/tex]
1. Maxima or Minima:
- [tex]\( g(x) \)[/tex] is a quadratic function (parabola) that opens downwards because the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\(-1\)[/tex]).
- A parabola that opens downwards has a maximum point (vertex).
- Conclusion: [tex]\( g(x) \)[/tex] does not have a minima. The correct interpretation is that [tex]\( g(x) \)[/tex] has a maxima, but here we focus on confirming that it does not have a minima (True if we consider maxima).
2. X-Intercept:
- To find the x-intercepts, set [tex]\( g(x) = 0 \)[/tex]:
[tex]\[
-x^2 + 9 = 0 \implies -x^2 = -9 \implies x^2 = 9 \implies x = \pm 3
\][/tex]
- Therefore, [tex]\( g(x) \)[/tex] intersects the x-axis at [tex]\( x = -3 \)[/tex] and [tex]\( x = 3 \)[/tex].
- Conclusion: [tex]\( g(x) \)[/tex] has two x-intercepts (False).
### Summary
- Statement: [tex]\( f(x) \)[/tex] has a maxima: FALSE
- Justification: Linear functions do not have maxima or minima.
- Statement: [tex]\( g(x) \)[/tex] has a minima: FALSE
- Correction: While the statement should have referred to the maxima, the analysis above indicates the function does not have a minima but has a maxima.
- Statement: Both functions have exactly one x-intercept: FALSE
- Justification: [tex]\( f(x) \)[/tex] has exactly one x-intercept at [tex]\( x = -3 \)[/tex], while [tex]\( g(x) \)[/tex] has two x-intercepts at [tex]\( x = -3 \)[/tex] and [tex]\( x = 3 \)[/tex].
### Conclusion
The statements must be carefully verified:
- [tex]\( f(x) \)[/tex] does not have a maxima (True if refined to be accurate).
- [tex]\( g(x) \)[/tex] does not have a minima (True).
- Both functions do not have exactly one x-intercept (True, since [tex]\( g(x) \)[/tex] has two).
