To determine which statement must be true regarding the functions \( f(x) \) and \( g(x) \) given that their sum, \( (f + g)(x) \), is a linear function represented by a straight line with a negative slope, we can analyze the implications:
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Quadratic Functions: Both \( f(x) \) and \( g(x) \) are quadratic functions, which means they can generally be expressed in the form: \[ f(x) = ax^2 + bx + c \] \[ g(x) = dx^2 + ex + k \]
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Sum of Quadratics: The sum \( (f + g)(x) = (a + d)x^2 + (b + e)x + (c + k) \).
- For \( (f + g)(x) \) to be linear, the coefficient of \( x^2 \) must be zero, which implies: \[ a + d = 0 \quad \Rightarrow \quad d = -a \]
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Interpretation of Leading Coefficients: From this conclusion, if \( f(x) \) has a leading coefficient \( a \) and \( g(x) \) has a leading coefficient \( d \), the condition \( d = -a \) tells us that the leading coefficients of \( f(x) \) and \( g(x) \) must be opposites.
Now let's evaluate the options provided:
- The leading coefficients of f(x) and g(x) are opposites. (TRUE, since \( d = -a \))
- The leading coefficients of f(x) and g(x) are opposite reciprocals. (FALSE, as they do not satisfy that condition)
- The leading coefficients of f(x) and g(x) are the same. (FALSE, since they are opposites)
- The leading coefficients of f(x) and g(x) are reciprocals. (FALSE, as they are not reciprocals)
Given the analysis, the statement that must be true is: The leading coefficients of f(x) and g(x) are opposites.