Answer :
Certainly! Let's go through the steps in detail.
### Step 1: Find the formula for [tex]\((f+g)(x)\)[/tex]
Given the functions:
[tex]\[ f(x) = -2x \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]
The combined function [tex]\((f+g)(x)\)[/tex] is defined as the sum of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
Now, substituting the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we get:
[tex]\[ (f + g)(x) = -2x + (x - 3) \][/tex]
### Simplify the formula for [tex]\((f+g)(x)\)[/tex]
Let's simplify this expression step by step:
[tex]\[ (f + g)(x) = -2x + x - 3 \][/tex]
Combine like terms:
[tex]\[ (f + g)(x) = -2x + x - 3 \][/tex]
[tex]\[ (f + g)(x) = -x - 3 \][/tex]
Thus, the simplified formula for [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ (f+g)(x) = -x - 3 \][/tex]
### Step 2: Find the domain of [tex]\((f+g)(x)\)[/tex]
The domain of a function is the set of all possible values of [tex]\(x\)[/tex] for which the function is defined.
### Analyzing domains of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
- The function [tex]\(f(x) = -2x\)[/tex] is defined for all real numbers, as there are no restrictions on [tex]\(x\)[/tex].
- The function [tex]\(g(x) = x - 3\)[/tex] is also defined for all real numbers, as there are no restrictions on [tex]\(x\)[/tex].
Since both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are defined for all real numbers, the domain of [tex]\((f+g)(x)\)[/tex] is also all real numbers.
### Conclusion:
The simplified formula for [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ (f+g)(x) = -x - 3 \][/tex]
The domain for [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ \text{all real numbers} \][/tex]
So, the final answers are:
- The formula for [tex]\((f+g)(x)\)[/tex] is: [tex]\[ -x - 3 \][/tex]
- The domain for [tex]\((f+g)(x)\)[/tex] is: [tex]\[ \text{all real numbers} \][/tex]
### Step 1: Find the formula for [tex]\((f+g)(x)\)[/tex]
Given the functions:
[tex]\[ f(x) = -2x \][/tex]
[tex]\[ g(x) = x - 3 \][/tex]
The combined function [tex]\((f+g)(x)\)[/tex] is defined as the sum of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
Now, substituting the expressions for [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex], we get:
[tex]\[ (f + g)(x) = -2x + (x - 3) \][/tex]
### Simplify the formula for [tex]\((f+g)(x)\)[/tex]
Let's simplify this expression step by step:
[tex]\[ (f + g)(x) = -2x + x - 3 \][/tex]
Combine like terms:
[tex]\[ (f + g)(x) = -2x + x - 3 \][/tex]
[tex]\[ (f + g)(x) = -x - 3 \][/tex]
Thus, the simplified formula for [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ (f+g)(x) = -x - 3 \][/tex]
### Step 2: Find the domain of [tex]\((f+g)(x)\)[/tex]
The domain of a function is the set of all possible values of [tex]\(x\)[/tex] for which the function is defined.
### Analyzing domains of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
- The function [tex]\(f(x) = -2x\)[/tex] is defined for all real numbers, as there are no restrictions on [tex]\(x\)[/tex].
- The function [tex]\(g(x) = x - 3\)[/tex] is also defined for all real numbers, as there are no restrictions on [tex]\(x\)[/tex].
Since both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex] are defined for all real numbers, the domain of [tex]\((f+g)(x)\)[/tex] is also all real numbers.
### Conclusion:
The simplified formula for [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ (f+g)(x) = -x - 3 \][/tex]
The domain for [tex]\((f+g)(x)\)[/tex] is:
[tex]\[ \text{all real numbers} \][/tex]
So, the final answers are:
- The formula for [tex]\((f+g)(x)\)[/tex] is: [tex]\[ -x - 3 \][/tex]
- The domain for [tex]\((f+g)(x)\)[/tex] is: [tex]\[ \text{all real numbers} \][/tex]
