91Ó°ÊÓ

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In function operations, algebra helps us combine and modify functions to obtain new ones. A function is a relationship between an input and an output, usually represented as \( f(x) \) or \( g(x) \). Function subtraction involves finding the difference between two functions' outputs for any given input value. If we have two functions, \( f(x) = x^2 + 2x - 7 \) and \( g(x) = 3x - 4 \), algebra allows us to manipulate these expressions to find their difference.

Function subtraction is straightforward once you understand it. To subtract one function from another, you follow these steps:

• Write down the expression for the difference: \[ (g - f)(x) = g(x) - f(x) \]
• Substitute the given functions into the expression: \[ (g - f)(x) = (3x - 4) - (x^2 + 2x - 7) \]
• Distribute the negative sign: \[ (g - f)(x) = 3x - 4 - x^2 - 2x + 7 \]
• Combine like terms: \[ (g - f)(x) = -x^2 + 3x - 2x + 7 - 4 \]
• Simplify: \[ (g - f)(x) = -x^2 + x + 3 \]

By following these steps, you can easily find the difference between any two functions.

Polynomial functions are a type of algebraic function and are expressed as a sum of terms, each term including a variable raised to a non-negative integer power and a coefficient. Examples include \( f(x) = x^2 + 2x - 7 \). In our problem, both \( f(x) \) and \( g(x) \) are polynomial functions. When manipulating polynomial functions, paying attention to the degree (the highest power of the variable) and combining like terms is crucial. Combining like terms involves adding or subtracting the coefficients of the same power variables and ensuring the final expression is in its simplest form. The goal is to reduce complex expressions into simpler polynomial forms like we did with \( -x^2 + x + 3 \).