Math Problem Statement

Consider the following functions: f(x) = 3x - 7, g(x) = 6 - x. Find (f + g)(x), (f - g)(x), (fg)(x), and (f/g)(x). What is the domain of (f/g)?

Solution

Let's break down the problems from your image and solve them step by step.

Given functions:

f(x)=3x7f(x) = 3x - 7
g(x)=6xg(x) = 6 - x

(a) Find (f+g)(x)(f + g)(x):

We add f(x)f(x) and g(x)g(x).

(f+g)(x)=f(x)+g(x)=(3x7)+(6x)=3xx7+6=2x1(f + g)(x) = f(x) + g(x) = (3x - 7) + (6 - x) = 3x - x - 7 + 6 = 2x - 1

So, (f+g)(x)=2x1(f + g)(x) = 2x - 1.

(b) Find (fg)(x)(f - g)(x):

We subtract g(x)g(x) from f(x)f(x).

(fg)(x)=f(x)g(x)=(3x7)(6x)=3x76+x=4x13(f - g)(x) = f(x) - g(x) = (3x - 7) - (6 - x) = 3x - 7 - 6 + x = 4x - 13

So, (fg)(x)=4x13(f - g)(x) = 4x - 13.

(c) Find (fg)(x)(fg)(x):

We multiply f(x)f(x) and g(x)g(x).

(fg)(x)=f(x)g(x)=(3x7)(6x)=3x(6x)7(6x)=18x3x242+7x=3x2+25x42(fg)(x) = f(x) \cdot g(x) = (3x - 7)(6 - x) = 3x(6 - x) - 7(6 - x) = 18x - 3x^2 - 42 + 7x = -3x^2 + 25x - 42

So, (fg)(x)=3x2+25x42(fg)(x) = -3x^2 + 25x - 42.

(d) Find (fg)(x)\left( \frac{f}{g} \right)(x):

We divide f(x)f(x) by g(x)g(x).

(fg)(x)=f(x)g(x)=3x76x\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{3x - 7}{6 - x}

Domain of fg\frac{f}{g}:

The domain of a rational function excludes values of xx that make the denominator zero. So, we solve:

6x=0    x=66 - x = 0 \implies x = 6

Thus, the domain is all real numbers except x=6x = 6. In interval notation, the domain is:

(,6)(6,)(-\infty, 6) \cup (6, \infty)

Summary of Answers:

(a) (f+g)(x)=2x1(f + g)(x) = 2x - 1
(b) (fg)(x)=4x13(f - g)(x) = 4x - 13
(c) (fg)(x)=3x2+25x42(fg)(x) = -3x^2 + 25x - 42
(d) (fg)(x)=3x76x\left( \frac{f}{g} \right)(x) = \frac{3x - 7}{6 - x}
Domain: (,6)(6,)(-\infty, 6) \cup (6, \infty)

Would you like further details on any part of these solutions?

Here are 5 related questions for practice:

  1. What is (g+f)(x)(g + f)(x) for different given functions?
  2. How would you find (gf)(x)(g - f)(x)?
  3. What happens to the domain if both f(x)f(x) and g(x)g(x) involve quadratic terms?
  4. How would you solve (gf)(x)(g \cdot f)(x) for more complex functions?
  5. What is the domain of fg\frac{f}{g} when both functions have multiple variables?

Tip: When finding domains of rational functions, always check when the denominator becomes zero.

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Math Problem Analysis

Mathematical Concepts

Algebra
Function Operations
Domain of Functions
Rational Functions

Formulas

(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(fg)(x) = f(x) * g(x)
(f/g)(x) = f(x) / g(x)

Theorems

Domain Exclusion Rule for Rational Functions

Suitable Grade Level

High School (Grades 9-12)