Subtracting Functions: Exploring f(x) - g(x)

In this article, we will explore the subtraction of two functions, f(x) and g(x), and determine the domain of the resulting function f(x) - g(x).

Let's start by defining the two functions:

1. f(x) = x^2 + 3x - 4
2. g(x) = (5x + 2) / (2x^2 - x - 1)

To find the function f(x) - g(x), we simply need to subtract the two functions:

f(x) - g(x) = (x^2 + 3x - 4) - (5x + 2) / (2x^2 - x - 1)

Simplifying the expression, we get:

f(x) - g(x) = (x^2 + 3x - 4) * (2x^2 - x - 1) - (5x + 2)

Multiplying out the numerator, we have:

f(x) - g(x) = 2x^4 + x^3 - 9x^2 - 7x - 4 - 5x - 2

Combining like terms, we get:

f(x) - g(x) = 2x^4 + x^3 - 9x^2 - 12x - 6

Now, let's determine the domain of the function f(x) - g(x).

The domain of f(x) is the set of all real numbers, as the function f(x) = x^2 + 3x - 4 is defined for all real values of x.

The domain of g(x) is the set of all real numbers x such that 2x^2 - x - 1 ≠ 0. Solving this inequality, we get:

x ≠ (1 ± √(1 + 8)) / 4

Therefore, the domain of g(x) is the set of all real numbers x except for (1 + √17) / 4 and (1 - √17) / 4.

The domain of the function f(x) - g(x) is the intersection of the domains of f(x) and g(x), which is the set of all real numbers x except for (1 + √17) / 4 and (1 - √17) / 4.

In summary, we have:

• f(x) = x^2 + 3x - 4
• g(x) = (5x + 2) / (2x^2 - x - 1)
• f(x) - g(x) = 2x^4 + x^3 - 9x^2 - 12x - 6
• The domain of f(x) - g(x) is the set of all real numbers x except for (1 + √17) / 4 and (1 - √17) / 4.