Math Problem Statement

Given f(x) = x^2 - 4 and g(x) = 3x + 1, find (f - g)(x).

Solution

The problem shown in the image defines two functions and asks for (fg)(x)(f - g)(x), where:

  • f(x)=x24f(x) = x^2 - 4
  • g(x)=3x+1g(x) = 3x + 1

To find (fg)(x)(f - g)(x), we calculate f(x)g(x)f(x) - g(x):

(fg)(x)=f(x)g(x)=(x24)(3x+1)(f - g)(x) = f(x) - g(x) = (x^2 - 4) - (3x + 1)

Now, distribute the subtraction:

(fg)(x)=x243x1(f - g)(x) = x^2 - 4 - 3x - 1

Combine like terms:

(fg)(x)=x23x5(f - g)(x) = x^2 - 3x - 5

Thus, the answer is:

(fg)(x)=x23x5(f - g)(x) = x^2 - 3x - 5

Would you like further details or have any questions?

Here are some related questions that expand on this topic:

  1. How would we find (f+g)(x)(f + g)(x) for the same functions?
  2. What is the derivative of (fg)(x)=x23x5(f - g)(x) = x^2 - 3x - 5?
  3. How do we calculate (fg)(x)(f \cdot g)(x) for f(x)=x24f(x) = x^2 - 4 and g(x)=3x+1g(x) = 3x + 1?
  4. What would be (f/g)(x)(f / g)(x) if g(x)0g(x) \neq 0?
  5. How can we determine the domain of (fg)(x)(f - g)(x)?

Tip: When subtracting functions, always distribute the subtraction carefully to avoid sign errors.

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Math Problem Analysis

Mathematical Concepts

Function Operations
Polynomial Subtraction

Formulas

(f - g)(x) = f(x) - g(x)

Theorems

Subtraction of Functions

Suitable Grade Level

Grades 9-10