Answer:
Function operation and function composition are two fundamental concepts in mathematics that are commonly used in algebra, calculus, and other branches of mathematics.
Function operation involves performing arithmetic operations on two or more functions to create a new function. To find the result of f(x) + g(x), we simply add the two functions together:
f(x) + g(x) = (2x - 1) + (x^2 - 9x - 4) = x^2 - 7x - 5
Similarly, we can find g(x) - f(x) and f(x) - g(x) by subtracting one function from the other:
g(x) - f(x) = (x^2 - 9x - 4) - (2x - 1) = x^2 - 11x - 3
f(x) - g(x) = (2x - 1) - (x^2 - 9x - 4) = -x^2 + 11x - 3
Function composition, on the other hand, involves plugging one function into another function to create a new function. To find g(f(x)), we first evaluate f(x) and then plug the result into g(x):
g(f(x)) = g(2x - 1) = (2x - 1)^2 - 9(2x - 1) - 4 = 4x^2 - 25x - 14
Similarly, we can find f(g(x)) by plugging g(x) into f(x):
f(g(x)) = f(x^2 - 9x - 4) = 2(x^2 - 9x - 4) - 1 = 2x^2 - 18x - 9
Now, to answer your question about why we don't need to do both f(x) + g(x) and g(x) + f(x), it's because addition is commutative, which means that the order of the terms doesn't matter. Therefore, f(x) + g(x) is the same as g(x) + f(x). The same is true for subtraction. However, this is not the case for function composition, as plugging one function into another is not commutative.