(1) If [tex]$f(x)=\sqrt{2 x-1}$[/tex] and [tex]$g(x)=1 / x$[/tex], find:
(a) [tex]$\left(\frac{f}{g}\right)(x)$[/tex]
(b) [tex]$(f+g)(x)$[/tex]
Answer (2)
To find ( g f ) ( x ) , divide f ( x ) by g ( x ) : x 1 2 x − 1 = x 2 x − 1 .
To find ( f + g ) ( x ) , add f ( x ) and g ( x ) : 2 x − 1 + x 1 .
The result of ( g f ) ( x ) is x 2 x − 1 .
The result of ( f + g ) ( x ) is 2 x − 1 + x 1 .
Explanation
Analysis of the problem We are given two functions, $f(x) =
!!
Finding (f/g)(x) (a) To find ( g f ) ( x ) , we need to divide f ( x ) by g ( x ) . Given f ( x ) = 2 x − 1 and g ( x ) = x 1 , we have: ( g f ) ( x ) = g ( x ) f ( x ) = x 1 2 x − 1 To simplify this expression, we multiply the numerator by the reciprocal of the denominator: ( g f ) ( x ) = x 2 x − 1
Finding (f+g)(x) (b) To find ( f + g ) ( x ) , we need to add f ( x ) and g ( x ) . Given f ( x ) = 2 x − 1 and g ( x ) = x 1 , we have: ( f + g ) ( x ) = f ( x ) + g ( x ) = 2 x − 1 + x 1
Final Answer Therefore, the final answers are: (a) ( g f ) ( x ) = x 2 x − 1 (b) ( f + g ) ( x ) = 2 x − 1 + x 1
Examples
Understanding function operations like division and addition is crucial in many real-world applications. For instance, consider a scenario where f ( x ) represents the production cost of x items and g ( x ) represents the number of items produced per hour. Then, g f ( x ) would give the cost per item per hour. Similarly, if f ( x ) represents the revenue from selling x items and g ( x ) represents the cost of marketing x items, then ( f + g ) ( x ) (although not directly interpretable) can be used in more complex models to analyze profitability by combining revenue and marketing costs. These operations help in optimizing business processes and resource allocation.
The function ( g f ) ( x ) is given by x 2 x − 1 , and the function ( f + g ) ( x ) is represented by 2 x − 1 + x 1 .
;
