Given the functions [tex]f(x)=-3 x+8[/tex] and [tex]g(x)=2 x+5[/tex], which operation, [tex]f+g[/tex], [tex]f-g[/tex], or [tex]f \cdot g[/tex], results in the smallest coefficient on the x term?
A. [tex]f \cdot g[/tex]
B. All operations result in the same coefficient
C. [tex]f-g[/tex]
D. [tex]f+g[/tex]
Answer (2)
Calculate f ( x ) + g ( x ) = − x + 13 , so the coefficient of x is -1.
Calculate f ( x ) − g ( x ) = − 5 x + 3 , so the coefficient of x is -5.
Calculate f ( x ) × g ( x ) = − 6 x 2 + x + 40 , so the coefficient of x is 1.
The smallest coefficient is -5, which comes from f ( x ) − g ( x ) .
Therefore, the answer is f − g .
Explanation
Problem Analysis We are given two functions, f ( x ) = − 3 x + 8 and g ( x ) = 2 x + 5 . Our goal is to determine which operation, f + g , f − g , or f × g , results in the smallest coefficient on the x term.
Calculate f(x) + g(x) First, let's calculate f ( x ) + g ( x ) : f ( x ) + g ( x ) = ( − 3 x + 8 ) + ( 2 x + 5 ) = − 3 x + 2 x + 8 + 5 = − x + 13
The coefficient of the x term in f ( x ) + g ( x ) is − 1 .
Calculate f(x) - g(x) Next, let's calculate f ( x ) − g ( x ) : f ( x ) − g ( x ) = ( − 3 x + 8 ) − ( 2 x + 5 ) = − 3 x − 2 x + 8 − 5 = − 5 x + 3 The coefficient of the x term in f ( x ) − g ( x ) is − 5 .
Calculate f(x) * g(x) Now, let's calculate f ( x ) × g ( x ) : f ( x ) × g ( x ) = ( − 3 x + 8 ) ( 2 x + 5 ) = − 3 x ( 2 x ) − 3 x ( 5 ) + 8 ( 2 x ) + 8 ( 5 ) = − 6 x 2 − 15 x + 16 x + 40 = − 6 x 2 + x + 40 The coefficient of the x term in f ( x ) × g ( x ) is 1 .
Compare Coefficients and Conclude Comparing the coefficients of the x terms, we have − 1 , − 5 , and 1 . The smallest of these is − 5 . Therefore, the operation f ( x ) − g ( x ) results in the smallest coefficient on the x term.
Examples
Understanding how coefficients change when combining functions is crucial in many real-world applications. For instance, in physics, if f ( x ) represents the force exerted by one object and g ( x ) represents the force exerted by another, the combined force might be f ( x ) + g ( x ) . Knowing how the coefficients combine helps predict the overall effect. Similarly, in economics, if f ( x ) and g ( x ) represent cost functions, understanding the resulting coefficients after operations like addition or subtraction can aid in budgeting and financial planning. This skill is also fundamental in engineering, where combining different signals or systems requires careful analysis of how their mathematical representations interact.
The operation f − g results in the smallest coefficient on the x-term, which is − 5 . In comparison, f + g has a coefficient of − 1 and f ⋅ g has a coefficient of 1 . Therefore, the correct answer is C. f − g .
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