Question

1. perform each operation and determine the domain of each function if: ( f(x) = x^2 - 3x - 4 ) and ( g(x) = x - 5 ) ( f(x) + g(x) = ) ( f(x) - g(x) = )

Explanation:

Step1: Substitute functions for sum

$f(x)+g(x) = (x^2 - 3x - 4) + (x - 5)$

Step2: Simplify the sum

$f(x)+g(x) = x^2 - 3x + x - 4 - 5 = x^2 - 2x - 9$

Step3: Find domain of sum

The resulting function is a polynomial, defined for all real numbers. Domain: $(-\infty, \infty)$

Step4: Substitute functions for difference

$f(x)-g(x) = (x^2 - 3x - 4) - (x - 5)$

Step5: Simplify the difference

$f(x)-g(x) = x^2 - 3x - 4 - x + 5 = x^2 - 4x + 1$

Step6: Find domain of difference

The resulting function is a polynomial, defined for all real numbers. Domain: $(-\infty, \infty)$

Answer:

$f(x)+g(x) = x^2 - 2x - 9$, Domain: $(-\infty, \infty)$
$f(x)-g(x) = x^2 - 4x + 1$, Domain: $(-\infty, \infty)$