For the pair of functions f(x) = √x and g(x) = x-11, find the following.
a. Find the functions f+g, f-g, fg, and f/g.
b. Determine the domain of the functions f+g, f-g, fg, and f/g.
Answer (2)
Find the sum of the functions: ( f + g ) ( x ) = 4 x + 11 x , domain: x ≥ 0 .
Find the difference of the functions: ( f − g ) ( x ) = 4 x − 11 x , domain: x ≥ 0 .
Find the product of the functions: ( f g ) ( x ) = 11 x 5/4 , domain: x ≥ 0 .
Find the quotient of the functions: ( g f ) ( x ) = 11 x 3/4 1 , domain: 0"> x > 0 .
( f + g ) ( x ) = 4 x + 11 x , ( f − g ) ( x ) = 4 x − 11 x , ( f g ) ( x ) = 11 x 5/4 , ( g f ) ( x ) = 11 x 3/4 1
Explanation
Understanding the Problem We are given two functions, f ( x ) = 4 x and g ( x ) = 11 x . Our goal is to find the functions f + g , f − g , f g , and g f , and then determine the domain of each of these new functions.
Finding f+g First, let's find f + g . This is simply the sum of the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) = 4 x + 11 x
Finding f-g Next, let's find f − g . This is the difference of the two functions: ( f − g ) ( x ) = f ( x ) − g ( x ) = 4 x − 11 x
Finding fg Now, let's find f g . This is the product of the two functions: ( f g ) ( x ) = f ( x ) g ( x ) = ( 4 x ) ( 11 x ) = 11 x 1/4 x 1 = 11 x 1/4 + 1 = 11 x 5/4
Finding f/g Finally, let's find g f . This is the quotient of the two functions: ( g f ) ( x ) = g ( x ) f ( x ) = 11 x 4 x = 11 x 1 x 1/4 = 11 1 x 1/4 − 1 = 11 1 x − 3/4 = 11 x 3/4 1 Note that x cannot be zero since it is in the denominator.
Determining the Domains Now, let's determine the domains of each of these functions. The domain of f ( x ) = 4 x is x ≥ 0 since we can only take the fourth root of non-negative numbers. The domain of g ( x ) = 11 x is all real numbers. For f + g and f − g , the domain is the intersection of the domains of f and g , which is x ≥ 0 .
For f g , the domain is also the intersection of the domains of f and g , which is x ≥ 0 .
For g f , we need to consider the domain of f , the domain of g , and also exclude any values of x where g ( x ) = 0 . Since g ( x ) = 11 x , g ( x ) = 0 when x = 0 . Therefore, the domain of g f is 0"> x > 0 .
Final Answer In summary: ( f + g ) ( x ) = 4 x + 11 x , Domain: x ≥ 0 ( f − g ) ( x ) = 4 x − 11 x , Domain: x ≥ 0 ( f g ) ( x ) = 11 x 5/4 , Domain: x ≥ 0 0"> ( g f ) ( x ) = 11 x 3/4 1 , Domain: x > 0
Examples
Understanding function operations and domains 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 revenue generated from selling x items. Then, ( f + g ) ( x ) could represent the total cost plus revenue, ( f − g ) ( x ) could represent the profit (if f(x)"> g ( x ) > f ( x ) ), ( f g ) ( x ) might represent a combined cost-revenue factor, and ( g f ) ( x ) could represent the cost per unit of revenue. Determining the domains ensures that these calculations are meaningful and realistic, as you can't produce or sell a negative number of items.
The functions are: f + g = x + x − 11 , f − g = x − x + 11 , f g = x 3/2 − 11 x , and g f = x − 11 x . The domains are as follows: f + g and f − g have domain x ≥ 0 , f g has domain x ≥ 0 , and g f has domain x ≥ 0 , x = 11 .
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