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)
We calculated the functions f + g , f − g , f g , and g f using the definitions of the functions. Their domains were determined, with f + g , f − g , and f g all having the domain [ 0 , ∞ ) , while g f has the domain [ 0 , 11 ) ∪ ( 11 , ∞ ) .
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Find the sum of the functions: ( f + g ) ( x ) = x + x − 11 with domain [ 0 , ∞ ) .
Find the difference of the functions: ( f − g ) ( x ) = x − x + 11 with domain [ 0 , ∞ ) .
Find the product of the functions: ( f g ) ( x ) = x ( x − 11 ) with domain [ 0 , ∞ ) .
Find the quotient of the functions: ( g f ) ( x ) = x − 11 x with domain [ 0 , 11 ) ∪ ( 11 , ∞ ) .
The functions and their domains are:
( f + g ) ( x ) = x + x − 11 , Domain: [ 0 , ∞ ) ( f − g ) ( x ) = x − x + 11 , Domain: [ 0 , ∞ ) ( f g ) ( x ) = x ( x − 11 ) , Domain: [ 0 , ∞ ) ( g f ) ( x ) = x − 11 x , Domain: [ 0 , 11 ) ∪ ( 11 , ∞ )
Explanation
Understanding the Problem We are given two functions, f ( x ) = x and g ( x ) = x − 11 . We need to find the expressions for f + g , f − g , f g , and g f , and then determine the domains of f + g , f − g , f g , and g f .
Finding f+g To find f + g , we add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) = x + x − 11.
Finding f-g To find f − g , we subtract g ( x ) from f ( x ) : ( f − g ) ( x ) = f ( x ) − g ( x ) = x − ( x − 11 ) = x − x + 11.
Finding fg To find f g , we multiply the two functions: ( f g ) ( x ) = f ( x ) ⋅ g ( x ) = x ( x − 11 ) .
Finding f/g To find g f , we divide f ( x ) by g ( x ) : ( g f ) ( x ) = g ( x ) f ( x ) = x − 11 x .
Analyzing Domains Now, let's determine the domains of these functions. The domain of f ( x ) = x is x ≥ 0 , since we cannot take the square root of a negative number. The domain of g ( x ) = x − 11 is all real numbers.
Domain of f+g, f-g, and fg For f + g , f − g , and f g , the domain is determined by the domain of f ( x ) , since g ( x ) is defined for all real numbers. Therefore, the domain of f + g , f − g , and f g is x ≥ 0 , or [ 0 , ∞ ) .
Domain of f/g For g f , we need to consider the domain of f ( x ) and also ensure that g ( x ) = 0 . We have x ≥ 0 and x − 11 = 0 , which means x = 11 . Therefore, the domain of g f is x ≥ 0 and x = 11 , or [ 0 , 11 ) ∪ ( 11 , ∞ ) .
Final Answer In summary:
( f + g ) ( x ) = x + x − 11 , Domain: [ 0 , ∞ ) ( f − g ) ( x ) = x − x + 11 , Domain: [ 0 , ∞ ) ( f g ) ( x ) = x ( x − 11 ) , Domain: [ 0 , ∞ ) ( g f ) ( x ) = x − 11 x , Domain: [ 0 , 11 ) ∪ ( 11 , ∞ )
Examples
Understanding function operations and their 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 financial outlay, ( f − g ) ( x ) could represent the profit (or loss), and g ( x ) f ( x ) could represent the cost-revenue ratio. Determining the domains ensures that we only consider realistic scenarios (e.g., non-negative production quantities) and avoid division by zero (e.g., when there is no revenue).
