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) $(f+g)(x)=$ $\square$ (Simplify your answer. Do not factor.)
Answer (1)
( f + g ) ( x ) = x 2 − 6 x + 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 ) , ( f "." g ) ( x ) , ( g f ) ( x ) , and the domain of g f .
Finding (f+g)(x) (a) To find ( f + g ) ( x ) , we add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) = ( x 2 − 7 x ) + ( 6 + x ) Combine like terms: ( f + g ) ( x ) = x 2 − 7 x + x + 6 = x 2 − 6 x + 6
Finding (f-g)(x) (b) 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 ) Distribute the negative sign and combine like terms: ( f − g ) ( x ) = x 2 − 7 x − 6 − x = x 2 − 8 x − 6
Finding (f \cdot g)(x) (c) To find ( f ⋅ g ) ( x ) , we multiply the two functions: ( f ⋅ g ) ( x ) = f ( x ) ⋅ g ( x ) = ( x 2 − 7 x ) ( 6 + x ) Expand the expression: ( f ⋅ g ) ( x ) = x 2 ( 6 + x ) − 7 x ( 6 + x ) = 6 x 2 + x 3 − 42 x − 7 x 2 Combine like terms: ( f ⋅ g ) ( x ) = x 3 − x 2 − 42 x
Finding (f/g)(x) (d) 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
Finding the domain of (f/g)(x) (e) To find the domain of g f , we need to find the values of x for which the denominator is not zero. So, we solve g ( x ) = 0 :
6 + x = 0 x = − 6 Therefore, the domain of g f is all real numbers except x = − 6 . In interval notation, this is ( − ∞ , − 6 ) ∪ ( − 6 , ∞ ) .
Final Answer (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 all real numbers except x = − 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. Similarly, in physics, if f ( x ) describes the distance traveled by an object and g ( x ) describes the time taken, then ( g f ) ( x ) gives the speed. Knowing the domain of these combined functions helps in understanding the limitations and realistic scenarios for these models.
