25. Given the functions [tex]$f(x) = -3x + 1$[/tex] and [tex]$g(x) = -2x - 3$[/tex], which operation results in the smallest coefficient on the [tex][tex]$x$[/tex][/tex] term?

A. [tex]f + g[/tex]
B. [tex]f \cdot g[/tex]
C. [tex]f - g[/tex]
D. All operations result in the same coefficient



Answer :

Let's analyze the functions and perform the three operations: addition, subtraction, and multiplication, focusing on the coefficients of the [tex]\(x\)[/tex] terms.

Given:
[tex]\[ f(x) = -3x + 1 \][/tex]
[tex]\[ g(x) = -2x - 3 \][/tex]

### 1. Addition of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
[tex]\[ (f + g)(x) = (-3x + 1) + (-2x - 3) \][/tex]
Combine like terms:
[tex]\[ (f + g)(x) = -3x - 2x + 1 - 3 \][/tex]
[tex]\[ (f + g)(x) = -5x - 2 \][/tex]
The coefficient of [tex]\(x\)[/tex] in [tex]\(f + g\)[/tex] is [tex]\(-5\)[/tex].

### 2. Subtraction of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f - g)(x) = f(x) - g(x) \][/tex]
[tex]\[ (f - g)(x) = (-3x + 1) - (-2x - 3) \][/tex]
Distribute the negative sign:
[tex]\[ (f - g)(x) = -3x + 1 + 2x + 3 \][/tex]
Combine like terms:
[tex]\[ (f - g)(x) = -3x + 2x + 1 + 3 \][/tex]
[tex]\[ (f - g)(x) = -x + 4 \][/tex]
The coefficient of [tex]\(x\)[/tex] in [tex]\(f - g\)[/tex] is [tex]\(-1\)[/tex].

### 3. Multiplication of [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[ (f \cdot g)(x) = f(x) \cdot g(x) \][/tex]
[tex]\[ (f \cdot g)(x) = (-3x + 1)(-2x - 3) \][/tex]
Multiply each term:
[tex]\[ (f \cdot g)(x) = (-3x)(-2x) + (-3x)(-3) + (1)(-2x) + (1)(-3) \][/tex]
[tex]\[ (f \cdot g)(x) = 6x^2 + 9x - 2x - 3 \][/tex]
Combine like terms:
[tex]\[ (f \cdot g)(x) = 6x^2 + 7x - 3 \][/tex]
The coefficient of [tex]\(x\)[/tex] (the linear term) in [tex]\(f \cdot g\)[/tex] is [tex]\(7\)[/tex].

### Summary of Coefficients
- For [tex]\(f + g\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(-5\)[/tex].
- For [tex]\(f - g\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(-1\)[/tex].
- For [tex]\(f \cdot g\)[/tex], the coefficient of [tex]\(x\)[/tex] is [tex]\(7\)[/tex].

### Conclusion
Among the coefficients [tex]\(-5\)[/tex], [tex]\(-1\)[/tex], and [tex]\(7\)[/tex], the smallest coefficient on the [tex]\(x\)[/tex] term is [tex]\(-5\)[/tex]. Therefore, the operation that results in the smallest coefficient on the [tex]\(x\)[/tex] term is:
[tex]\[ f + g \][/tex]

So, the correct answer is:
[tex]\[ \boxed{f+g} \][/tex]