Let $f(x)=x^2-7 x$ and $g(x)=6+x$. Find the following.
(a) $( f + g )( x )$
(b) $( f - g )( x )$
(c) $( f \cdot g )( x )$
(d) $\left(\frac{f}{g}\right)(x)$
(e) The domain of $\frac{f}{g}$
(a) $(f+g)(x)=x^2-6 x+6$ (Simplify your answer. Do not factor.)
(b) $(f-g)(x)=$ $\square$ (Simplify your answer. Do not factor.)
Answer (1)
Subtract g(x) from f(x) to find (f-g)(x): ( f − g ) ( x ) = ( x 2 − 7 x ) − ( 6 + x ) = x 2 − 8 x − 6 .
Multiply f(x) and g(x) to find (f \cdot g)(x): ( f ⋅ g ) ( x ) = ( x 2 − 7 x ) ( 6 + x ) = x 3 − x 2 − 42 x .
Divide f(x) by g(x) to find (f/g)(x): ( g f ) ( x ) = 6 + x x 2 − 7 x .
Find the domain of (f/g)(x) by setting the denominator 6 + x = 0 , which gives x = − 6 . The domain is ( − ∞ , − 6 ) ∪ ( − 6 , ∞ ) .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x 2 − 7 x and g ( x ) = 6 + x . We need to find ( f − g ) ( x ) , ( f "." g ) ( x ) , ( g f ) ( x ) , and the domain of g f . Part (a), ( f + g ) ( x ) , is already solved.
Finding (f-g)(x) To find ( f − g ) ( x ) , we subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) = ( x 2 − 7 x ) − ( 6 + x ) Simplifying the expression, we get: ( f − g ) ( x ) = x 2 − 7 x − 6 − x = x 2 − 8 x − 6 So, ( f − g ) ( x ) = x 2 − 8 x − 6 .
Finding (f \cdot g)(x) To find ( f ⋅ g ) ( x ) , we multiply f ( x ) and g ( x ) : ( f ⋅ g ) ( x ) = f ( x ) ⋅ g ( x ) = ( x 2 − 7 x ) ( 6 + x ) Expanding the expression, we get: ( f ⋅ g ) ( x ) = x 2 ( 6 + x ) − 7 x ( 6 + x ) = 6 x 2 + x 3 − 42 x − 7 x 2 = x 3 − x 2 − 42 x So, ( f ⋅ g ) ( x ) = x 3 − x 2 − 42 x .
Finding (f/g)(x) To find ( g f ) ( x ) , we divide f ( x ) by g ( x ) : ( g f ) ( x ) = g ( x ) f ( x ) = 6 + x x 2 − 7 x So, ( g f ) ( x ) = 6 + x x 2 − 7 x .
Finding the Domain of (f/g)(x) To find the domain of ( g f ) ( x ) , we need to find the values of x for which the denominator g ( x ) is not zero. We solve g ( x ) = 6 + x = 0 for x :
6 + x = 0 ⟹ x = − 6 Thus, the domain of g f is all real numbers except x = − 6 . In interval notation, the domain is ( − ∞ , − 6 ) ∪ ( − 6 , ∞ ) .
Final Answer Therefore, we have: (a) ( f + g ) ( x ) = x 2 − 6 x + 6 (b) ( f − g ) ( x ) = x 2 − 8 x − 6 (c) ( f ⋅ g ) ( x ) = x 3 − x 2 − 42 x (d) ( g f ) ( x ) = 6 + x x 2 − 7 x (e) The domain of g f is ( − ∞ , − 6 ) ∪ ( − 6 , ∞ ) .
Examples
Understanding function operations like addition, subtraction, multiplication, and division is crucial in many real-world applications. For instance, in economics, if f ( x ) represents the revenue of a company and g ( x ) represents the cost, then ( f − g ) ( x ) gives the profit. The domain of g f is important because it tells us for which values of x the profit calculation is valid. Similarly, in physics, if f ( x ) represents the distance traveled and g ( x ) represents the time taken, then g f gives the speed, and understanding its domain helps in analyzing the motion.
