Alright, let's analyze the given functions step-by-step and verify their domains to ensure if the domains indeed cover all real numbers:
We have two functions:
[tex]\( f(x) = 3x + 9 \)[/tex]
[tex]\( g(x) = -x^2 + 9 \)[/tex]
### Step-by-Step Analysis:
#### Function 1: [tex]\( f(x) = 3x + 9 \)[/tex]
1. Form: This is a linear function.
2. Domain: For linear functions, there are no restrictions on the values that [tex]\( x \)[/tex] can take. This means [tex]\( x \)[/tex] can be any real number.
3. Conclusion: The domain of [tex]\( f(x) = 3x + 9 \)[/tex] is all real numbers.
#### Function 2: [tex]\( g(x) = -x^2 + 9 \)[/tex]
1. Form: This is a quadratic function in the form of [tex]\( -x^2 + 9 \)[/tex].
2. Domain: Quadratic functions also have domains that include all real numbers, as there are no restrictions that would prevent [tex]\( x \)[/tex] from taking any real value.
3. Conclusion: The domain of [tex]\( g(x) = -x^2 + 9 \)[/tex] is all real numbers.
### Justification:
Both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are indeed defined for all real numbers. There are no restrictions or conditions that limit [tex]\( x \)[/tex] in either function.
Thus, the statements:
1. The domain of [tex]\( f(x) = 3x + 9 \)[/tex] is all real numbers.
2. The domain of [tex]\( g(x) = -x^2 + 9 \)[/tex] is all real numbers.
are TRUE.
As a result, the correct marking for both functions being defined for all real numbers is:
✓ TRUE
✓ TRUE
