To solve this problem, we need to describe the functions \( f(x) \) and \( g(x) \) given the conditions that their sum and product must match specific forms:
1. Sum Condition:
[tex]\[ h(x) = f(x) + g(x) = -x + 9 \][/tex]
2. Product Condition:
[tex]\[ j(x) = f(x) \cdot g(x) = -9x \][/tex]
We are given five statements to consider. We need to select the two correct ones that describe \( f(x) \) and \( g(x) \).
1. Both functions must be quadratic.:
- This is not necessarily true since linear functions can also satisfy the given conditions.
2. Both functions must have a constant rate of change.:
- This statement means both \( f(x) \) and \( g(x) \) are linear functions. The sum and product conditions can be met if \( f(x) \) and \( g(x) \) are linear. Thus, this statement is true.
3. Both functions must have a \( y \)-intercept of 0.:
- This statement means that each function must pass through the origin (i.e., when \( x = 0 \), \( f(0) = 0 \) and \( g(0) = 0 \)). Given the constraints, both functions can indeed pass through the origin. Hence, \( f(x) \) and \( g(x) \) can have a \( y \)-intercept of 0. So, this seem true.
4. The rate of change of either \( f(x) \) or \( g(x) \) must be 0.:
- This means one of the functions is a constant function and the other is linear. This does not fit with the requirement that they have a sum forming a linear equation with a nonzero slope. Thus, this statement is false.
5. The \( y \)-intercepts of \( f(x) \) and \( g(x) \) must be opposites.:
- If the \( y \)-intercepts had to be opposites, then \( f(x) + g(x) \) would not have a constant term as \( b + (-b) = 0 \). Therefore, this statement is false.
Based on the analysis:
- Both functions must have a constant rate of change.
- Both functions must have a \( y \)-intercept of 0.
So, the correct options are:
- Both functions must have a constant rate of change.
- Both functions must have a [tex]\( y \)[/tex]-intercept of 0.
