Are These Functions Equivalent: A Detailed Analysis of fx and gx

When dealing with polynomial functions, it is often necessary to check if they are equivalent. In this analysis, we will explore two functions, f(x) 5x^2 - 3x - 8 - x^2 and g(x) -3x - 2x^1 7, and determine if they are indeed equivalent. This involves simplifying both functions, comparing their terms, and evaluating them for specific values of x.

Simplifying Function fx

Step 1: Initial Equation - We start with the function f(x) 5x^2 - 3x - 8 - x^2. Our goal is to simplify it by combining like terms.

Step 2: Grouping and Simplifying - First, we combine the terms with the highest degree, which in this case are the terms containing x^2.

1. Combine the terms with x^2: (5x^2) - (x^2) 4x^2

2. Keep the linear term: -3x

3. Combine the constant terms: -8 - 1 -9

Thus, the simplified form of f(x) is 4x^2 - 3x - 9.

Expanding and Simplifying Function gx

Step 1: Initial Equation - The function g(x) -3x - 2x^1 7 is already in a form that looks simpler, so we will simplify it by combining like terms.

Step 2: Grouping and Simplifying - Here, we need to combine the terms containing x.

1. Combine the terms with x: -3x - 2x -5x

2. Keep the constant term: 7

Thus, the simplified form of g(x) is 5x^2 - 5x 7.

Comparing the Simplified Functions

Step 1: Inspection - Upon inspection, we see that the two simplified forms of the functions are different. fx simplifies to 4x^2 - 3x - 9, and gx simplifies to 5x^2 - 5x 7. This indicates that the two functions are not equivalent.

Step 2: Further Check - To further verify, we can evaluate both functions at a specific value of x, such as x 1.

1. Evaluate f(1): 4(1)^2 - 3(1) - 9 4 - 3 - 9 -8

2. Evaluate g(1): 5(1)^2 - 5(1) 7 5 - 5 7 7

Clearly, f(1) -8 and g(1) 7, which are not the same. This confirms that the functions are not equivalent.

Conclusion

From our detailed analysis, it is clear that the functions f(x) 5x^2 - 3x - 8 - x^2 and g(x) -3x - 2x^1 7 are not equivalent. The differences lie in the coefficients of the highest degree term, x^2. fx simplifies to 4x^2 - 3x - 9, while gx simplifies to 5x^2 - 5x 7. These differences are significant and can be observed through both examination and specific evaluation at a value of x.

Understanding these concepts is crucial in algebra and polynomial functions, as it helps in identifying equivalent expressions and simplifying complex polynomials.

Keywords: polynomial functions, algebraic simplification, equivalent functions