For the functions 
f
 and 
g
:

f
⁡
(
x
)
=
3
⁢
x
-
2
   ;   
g
⁡
(
x
)
=
2
⁢
x
-
1


find 
(
f
+
g
)
⁢
(
x
)
 and its domain.

(
f
+
g
)
⁢
(
x
)
=
   

Domain (in interval notation):   

To find \((f + g)(x)\), we add the two functions \(f(x)\) and \(g(x)\) together:

### Given:
\[
f(x) = 3x - 2
\]
\[
g(x) = 2x - 1
\]

### Step 1: Add the two functions
\[
(f + g)(x) = f(x) + g(x) = (3x - 2) + (2x - 1)
\]

### Step 2: Simplify the expression
\[
(f + g)(x) = 3x + 2x - 2 - 1 = 5x - 3
\]

So, \((f + g)(x) = 5x - 3\).

### Step 3: Domain
Since both \(f(x)\) and \(g(x)\) are linear functions (polynomials of degree 1), their domains are all real numbers, \(\mathbb{R}\). Adding them together does not restrict the domain.

Thus, the domain of \((f + g)(x)\) is all real numbers, written in interval notation as:

\[
\text{Domain} = (-\infty, \infty)
\]

### Final Answer:
\[
(f + g)(x) = 5x - 3
\]
\[
\text{Domain} = (-\infty, \infty)
\]

Do you want more details on how the domain was determined or any other questions?

---

Here are 5 related questions to further explore this topic:
1. What is \((f - g)(x)\) for the given functions?
2. What would \((f \cdot g)(x)\) look like?
3. Can you find the composite function \(f(g(x))\)?
4. How would you find the inverse of \(f(x)\)?
5. What is the slope of the line represented by \(f(x)\)?

**Tip:** When dealing with basic polynomial functions, the domain is often all real numbers unless you encounter divisions by zero or square roots of negative numbers.

For the functions f(x) = 3x - 2 and g(x) = 2x - 1, find (f + g)(x) and its domain.

Learn how to add the functions f(x) = 3x - 2 and g(x) = 2x - 1 to find (f + g)(x) and its domain. This problem covers algebraic concepts, including function addition and domains of linear functions. Suitable for students in Grades 9-10.

Math Problem Statement

f ⁡ ( x )

3 ⁢ x

2 ;
g ⁡ ( x )

2 ⁢ x

( f + g ) ⁢ ( x )

Solution

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