Let $f(x)=x^2-7x$ 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 $\frac{f}{g}$
Answer (2)
Multiply f ( x ) and g ( x ) to find ( f \timges g ) ( x ) : ( x 2 − 7 x ) ( 6 + x ) = x 3 − x 2 − 42 x .
Divide f ( x ) by g ( x ) to find ( g f ) ( x ) : 6 + x x 2 − 7 x .
Solve g ( x ) = 0 to find the value(s) excluded from the domain of ( g f ) ( x ) : 6 + x = 0 ⟹ x = − 6 .
State the domain of ( g f ) ( x ) as all real numbers except x = − 6 .
The final answer is ( f \timges g ) ( x ) = x 3 − x 2 − 42 x .
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 \timges g ) ( x ) , ( g f ) ( x ) , and the domain of g f . Parts (a) and (b) are already given: ( f + g ) ( x ) = x 2 − 6 x + 6 and ( f − g ) ( x ) = x 2 − 8 x − 6 .
Calculating (f * g)(x) To find ( f \timges g ) ( x ) , we need to multiply the two functions: ( f \timges g ) ( x ) = f ( x ) \timges g ( x ) = ( x 2 − 7 x ) ( 6 + x ) Expanding this product, we get: ( f \timges 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 \timges g ) ( x ) = x 3 − x 2 − 42 x .
Calculating (f/g)(x) To find ( g f ) ( x ) , we need to divide f ( x ) by g ( x ) : ( g f ) ( x ) = g ( x ) f ( x ) = 6 + x x 2 − 7 x This expression is already simplified, 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 ) = 6 + x is not equal to zero. So, we need to solve the equation 6 + x = 0 :
6 + x = 0 ⟹ x = − 6 Therefore, the domain of ( g f ) ( x ) is all real numbers except x = − 6 . In interval notation, the domain is ( − ∞ , − 6 ) ∪ ( − 6 , ∞ ) .
Final Answer In conclusion: (a) ( f + g ) ( x ) = x 2 − 6 x + 6 (b) ( f − g ) ( x ) = x 2 − 8 x − 6 (c) ( f \timges 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 cost of producing x items and g ( x ) represents the revenue from selling x items, then ( f − g ) ( x ) represents the profit. Analyzing the domain of ( g f ) ( x ) helps determine for what quantities the revenue is non-zero, which is essential for business planning. These concepts are also used in physics to combine different forces or potentials and in computer graphics to manipulate images and animations.
The addition of the functions is ( f + g ) ( x ) = x 2 − 6 x + 6 , the subtraction results in ( f − g ) ( x ) = x 2 − 8 x − 6 , the product gives ( f × g ) ( x ) = x 3 − x 2 − 42 x , and the quotient is ( g f ) ( x ) = 6 + x x 2 − 7 x with the domain of g f being all real numbers except x = − 6 .
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