Perform the indicated operations on the functions given below.

A. Let f(x) = 5x + 1, g(x) = 5x - 1, and h(x) = 10x² + 13x - 3.

1. f(x) + g(x)
Solution:

2. h(x) - g(x)
Solution:

3. f(x)g(x)
Solution:

4. h(x) / g(x)
Solution:

5. g(x) / h(x)
Solution:

We performed five operations on the functions f(x), g(x), and h(x), resulting in f(x) + g(x) = 10x, h(x) - g(x) = 10x² + 8x - 2, f(x)g(x) = 25x² - 1, g ( x ) h ( x ) ​ = 5 x − 1 10 x 2 + 13 x − 3 ​ , and h ( x ) g ( x ) ​ = 10 x 2 + 13 x − 3 5 x − 1 ​ . Each operation utilized basic principles of algebra, such as combining like terms and applying the difference of squares. Finally, some expressions were left in their fractional forms as they could not be simplified further without specific values.
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Answered by Anonymous | 2025-07-04

Let's perform the operations on the functions given:
A. Let f ( x ) = 5 x + 1 , g ( x ) = 5 x − 1 , and h ( x ) = 10 x 2 + 13 x − 3 .

f ( x ) + g ( x )
To add f ( x ) and g ( x ) , add their expressions:


f ( x ) + g ( x ) = ( 5 x + 1 ) + ( 5 x − 1 ) = 5 x + 5 x + 1 − 1 = 10 x So, f ( x ) + g ( x ) = 10 x .

h ( x ) − g ( x )
To subtract g ( x ) from h ( x ) , subtract their expressions:


h ( x ) − g ( x ) = ( 10 x 2 + 13 x − 3 ) − ( 5 x − 1 ) Distribute the minus sign and simplify: = 10 x 2 + 13 x − 3 − 5 x + 1 = 10 x 2 + ( 13 x − 5 x ) + ( − 3 + 1 ) = 10 x 2 + 8 x − 2 Thus, h ( x ) − g ( x ) = 10 x 2 + 8 x − 2 .

f ( x ) g ( x )
Multiply f ( x ) and g ( x ) :


f ( x ) g ( x ) = ( 5 x + 1 ) ( 5 x − 1 ) This is a difference of squares: = 5 x ⋅ 5 x − 1 ⋅ 1 = 25 x 2 − 1 Therefore, f ( x ) g ( x ) = 25 x 2 − 1 .

g ( x ) h ( x ) ​
To divide h ( x ) by g ( x ) , we write:


g ( x ) h ( x ) ​ = 5 x − 1 10 x 2 + 13 x − 3 ​ Performing polynomial long division or synthetic division would provide the quotient and remainder, but since this expression might not simplify easily, you may need computational tools or manual calculation to see if there's a simpler form.

h ( x ) g ( x ) ​
To divide g ( x ) by h ( x ) , we write:


h ( x ) g ( x ) ​ = 10 x 2 + 13 x − 3 5 x − 1 ​ This expression is already in its simplest form unless additional information is given for specific values of x or further simplification techniques are employed.
These operations illustrate how algebraic expressions can be combined and manipulated through addition, subtraction, multiplication, and division.

Answered by OliviaLunaGracy | 2025-07-06