https://file.aiquickdraw.com/mathbot/file/1725502515533dsxylqoh.jpg

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.

If f(x) = x^2 - x + 9 and g(x) = 4x - 3, find (a) (f + g)(x) and (b) (f - g)(x), along with their domains.

Learn how to find the sum and difference of two polynomial functions f(x) = x^2 - x + 9 and g(x) = 4x - 3, and understand their domains. This problem focuses on polynomial function operations and domain determination. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation