Answer:
The domain of [tex]((f-g)(x) = 3x^2 - 3x + 1)[/tex] is all real numbers, which can be written in interval notation as ((-∞, ∞)).
Step-by-step explanation:
To find [tex]((f-g)(x))[/tex], we need to subtract the function [tex](g(x))[/tex] from [tex](f(x))[/tex]. Let's start by writing down the functions given:
[tex][f(x) = 3x^2 + 2x] [g(x) = 5x - 1][/tex]
Now, [tex]((f-g)(x))[/tex] means we subtract (g(x)) from [tex](f(x))[/tex]:
[tex][(f-g)(x) = f(x) - g(x)][/tex]
Substitute the given functions into this formula:
[tex][(f-g)(x) = (3x^2 + 2x) - (5x - 1)][/tex]
Now, simplify the expression by distributing the negative sign through the second parentheses:
[tex][(f-g)(x) = 3x^2 + 2x - 5x + 1][/tex]
Combine like terms:
[tex][(f-g)(x) = 3x^2 - 3x + 1][/tex]
So, [tex]((f-g)(x) = 3x^2 - 3x + 1)[/tex]
Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
The function [tex](f(x) = 3x^2 + 2x)[/tex] is a polynomial function, and polynomial functions have a domain of all real numbers, because you can plug any real number into (x) and get a real number out.
The function [tex](g(x) = 5x - 1)[/tex] is also a polynomial function and likewise has a domain of all real numbers for the same reason.
When we subtract these two functions to get [tex](3x^2 - 3x + 1)[/tex], the result is still a polynomial function. Polynomial functions do not have restrictions on their domain because you can input any real number into them.
Therefore, the domain of [tex]((f-g)(x) = 3x^2 - 3x + 1)[/tex] is all real numbers, which can be written in interval notation as ((-∞, ∞)).
Hope this helps!