Subtracting Functions: f(x) - g(x)
As high school mathematics teachers, we often encounter situations where students need to understand the operations between different functions. In this article, we will explore the subtraction of two functions, f(x) = x^2 + 3x - 4 and g(x) = (5x + 2)/(2x^2 - x - 1), and determine the domain of the resulting function.
Step 1: Understand the Functions
Let's start by understanding the individual functions:
f(x) = x^2 + 3x - 4: This is a quadratic function with a parabolic shape. The domain of this function is the set of all real numbers, as there are no restrictions on the input variablex.g(x) = (5x + 2)/(2x^2 - x - 1): This is a rational function, which is the ratio of two polynomial functions. The domain of this function is the set of all real numbersxsuch that2x^2 - x - 1 ≠ 0.
Step 2: Subtract the Functions
To subtract the functions, we simply need to subtract the corresponding function values:
(f - g)(x) = f(x) - g(x)
Substituting the given functions, we get:
(f - g)(x) = (x^2 + 3x - 4) - (5x + 2)/(2x^2 - x - 1)
Simplifying the expression, we get:
(f - g)(x) = (x^2 + 3x - 4) - (5x + 2)/(2x^2 - x - 1)
Step 3: Determine the Domain of (f - g)(x)
The domain of the difference function (f - g)(x) is the set of all real numbers x for which both f(x) and g(x) are defined.
The domain of f(x) is the set of all real numbers, as there are no restrictions on the input variable x.
The domain of g(x) is the set of all real numbers x such that 2x^2 - x - 1 ≠ 0. This means that the domain of (f - g)(x) is the set of all real numbers x that satisfy the condition 2x^2 - x - 1 ≠ 0.
Therefore, the domain of (f - g)(x) is the set of all real numbers x such that 2x^2 - x - 1 ≠ 0.
In conclusion, we have explored the subtraction of two functions, f(x) = x^2 + 3x - 4 and g(x) = (5x + 2)/(2x^2 - x - 1), and determined the domain of the resulting function (f - g)(x). This understanding of function operations and their domains is essential for students in high school mathematics.