Let's solve the problems shown in the image step-by-step.

Given Functions:

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

Problem (a): Find $(f + g)(x)$ and its domain

Solution:

$(f + g)(x) = f(x) + g(x)$

Substituting the given functions:

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

Simplify the expression:

$(f + g)(x) = x^2 + 3x + 6$

Domain: Both $f(x)$ and $g(x)$ are polynomials, which are defined for all real numbers. Thus, the domain is:

$\text{Domain} = (-\infty, \infty)$

Problem (b): Find $(f - g)(x)$ and its domain

Solution:

$(f - g)(x) = f(x) - g(x)$

Substituting the given functions:

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

Simplify the expression:

$(f - g)(x) = x^2 - 5x + 12$

Domain: Again, since both $f(x)$ and $g(x)$ are polynomials, their difference is defined for all real numbers:

$\text{Domain} = (-\infty, \infty)$

Summary:

1. $(f + g)(x) = x^2 + 3x + 6$ with Domain = $(- \infty, \infty)$
2. $(f - g)(x) = x^2 - 5x + 12$ with Domain = $(- \infty, \infty)$

Would you like more details on any of these steps or have any questions?

Related Questions:

1. How do you find the sum of two functions?
2. What are the steps to determine the domain of a function?
3. How do you simplify algebraic expressions involving polynomials?
4. How would the domain change if one function involved a square root?
5. How do you verify the domain of a combined function?

Tip:

Always check the individual domains of functions before combining them, especially when dealing with fractions or square roots that could restrict the domain.