For the pair of functions [tex]f(x)=5 x+4[/tex] and [tex]g(x)=x^2-1[/tex] find the following.
a. Find the functions [tex]f+g, f-g, fg[/tex], and [tex]\frac{f}{g}[/tex].
b. Determine the domain of the functions [tex]f+g, f-g, fg[/tex], and [tex]\frac{f}{g}[/tex].

Find the sum of the functions: ( f + g ) ( x ) = f ( x ) + g ( x ) = x 2 + 5 x + 3 .
Find the difference of the functions: ( f − g ) ( x ) = f ( x ) − g ( x ) = − x 2 + 5 x + 5 .
Find the product of the functions: ( f g ) ( x ) = f ( x ) ⋅ g ( x ) = ( 5 x + 4 ) ( x 2 − 1 ) = 5 x 3 + 4 x 2 − 5 x − 4 .
Find the quotient of the functions: ( g f ​ ) ( x ) = g ( x ) f ( x ) ​ = x 2 − 1 5 x + 4 ​ , and determine the domain restrictions where x  = ± 1 , so the domain is ( − ∞ , − 1 ) ∪ ( − 1 , 1 ) ∪ ( 1 , ∞ ) .
The functions f + g , f − g , and f g have a domain of all real numbers, ( − ∞ , ∞ ) .
The final answers are: ( f + g ) ( x ) = x 2 + 5 x + 3 ​ , ( f − g ) ( x ) = − x 2 + 5 x + 5 ​ , ( f g ) ( x ) = 5 x 3 + 4 x 2 − 5 x − 4 ​ , ( g f ​ ) ( x ) = x 2 − 1 5 x + 4 ​ ​ with domain ( − ∞ , − 1 ) ∪ ( − 1 , 1 ) ∪ ( 1 , ∞ ) .

Explanation

Understanding the Problem We are given two functions, f ( x ) = 5 x + 4 and g ( x ) = x 2 − 1 . Our goal is to find the expressions for f + g , f − g , f g , and g f ​ , and then determine the domain of each of these new functions.

Finding f+g To find f + g , we add the two functions: ( f + g ) ( x ) = f ( x ) + g ( x ) = ( 5 x + 4 ) + ( x 2 − 1 ) = x 2 + 5 x + 3.

Finding f-g To find f − g , we subtract g ( x ) from f ( x ) :
( f − g ) ( x ) = f ( x ) − g ( x ) = ( 5 x + 4 ) − ( x 2 − 1 ) = 5 x + 4 − x 2 + 1 = − x 2 + 5 x + 5.

Finding fg To find f g , we multiply the two functions: ( f g ) ( x ) = f ( x ) g ( x ) = ( 5 x + 4 ) ( x 2 − 1 ) = 5 x 3 + 4 x 2 − 5 x − 4.

Finding f/g To find g f ​ , we divide f ( x ) by g ( x ) :
( g f ​ ) ( x ) = g ( x ) f ( x ) ​ = x 2 − 1 5 x + 4 ​ .

Determining the Domains Now, let's determine the domains of each of these functions.


For f + g , f − g , and f g , since both f ( x ) and g ( x ) are polynomials, they are defined for all real numbers. Therefore, the domains of f + g , f − g , and f g are all real numbers, which can be written as ( − ∞ , ∞ ) .
For g f ​ , we need to find the values of x for which g ( x ) = 0 , since division by zero is undefined. We have g ( x ) = x 2 − 1 = ( x − 1 ) ( x + 1 ) . Thus, g ( x ) = 0 when x = 1 or x = − 1 . Therefore, the domain of g f ​ is all real numbers except x = 1 and x = − 1 . In interval notation, this is ( − ∞ , − 1 ) ∪ ( − 1 , 1 ) ∪ ( 1 , ∞ ) .

Final Answer In summary:


( f + g ) ( x ) = x 2 + 5 x + 3 , Domain: ( − ∞ , ∞ )
( f − g ) ( x ) = − x 2 + 5 x + 5 , Domain: ( − ∞ , ∞ )
( f g ) ( x ) = 5 x 3 + 4 x 2 − 5 x − 4 , Domain: ( − ∞ , ∞ )
( g f ​ ) ( x ) = x 2 − 1 5 x + 4 ​ , Domain: ( − ∞ , − 1 ) ∪ ( − 1 , 1 ) ∪ ( 1 , ∞ )

Examples
Understanding function operations and their domains is crucial in many real-world applications. For instance, in physics, if f ( x ) represents the distance an object travels over time and g ( x ) represents its velocity, then ( f + g ) ( x ) could describe the combined effect of distance and velocity on another related quantity. Similarly, in economics, if f ( x ) is the revenue function and g ( x ) is the cost function, then ( f − g ) ( x ) represents the profit function. The domain of these functions is important because it tells us the range of input values for which the model is valid, such as time intervals or production levels.

Answered by GinnyAnswer | 2025-07-03

We calculated sums, differences, products, and quotients of two functions f ( x ) and g ( x ) . The domains for the sum, difference, and product are all real numbers, while the quotient's domain excludes x = 1 and x = − 1 . The final answers are summarized accordingly.
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Answered by Anonymous | 2025-07-04