Question

mathematics 30-1 2 assignment boo
(1 mark) 2. is f(x) + g(x) = g(x) + f(x) true for all functions?
if not, what are the conditions required for the statement to be true?
expectations
writing one example will not justify the statement is true for all
functions. to prove it is true for all functions, students need to
describe the operation property found in the moodle lessons to support
the validity of their conclusion.
to prove the statement is false, students only need to provide a counterexample

Explanation:

Step1: Recall the commutative property of addition

The commutative property of addition states that for any two numbers \(a\) and \(b\), \(a + b=b + a\). When we consider functions \(f(x)\) and \(g(x)\), the sum \(f(x)+g(x)\) involves adding the output values of the functions for a given input \(x\).

Step2: Analyze the addition of function outputs

Let's take an arbitrary input \(x\). Let \(f(x)\) be a function and \(g(x)\) be another function. The value of \(f(x)+g(x)\) is the sum of the two real - valued (assuming the functions are real - valued, which is a common case) outputs of the functions at \(x\). Since addition of real numbers is commutative, for any \(x\) in the domain of both \(f\) and \(g\), \(f(x)+g(x)=g(x)+f(x)\). This holds for all functions \(f\) and \(g\) as long as the domains of \(f\) and \(g\) are such that the sum \(f(x)+g(x)\) is defined (i.e., the domains of \(f\) and \(g\) overlap enough so that for the values of \(x\) we consider, both \(f(x)\) and \(g(x)\) are defined and their sum is a valid output in the codomain). In the context of functions, when we talk about \(f(x)+g(x)\) and \(g(x)+f(x)\), we are essentially adding two quantities (the outputs of the functions) at each point \(x\) in the common domain of \(f\) and \(g\). And since addition of the quantities (numbers) is commutative, the equation \(f(x)+g(x)=g(x)+f(x)\) is true for all functions \(f\) and \(g\) for which the sum is defined (i.e., for all \(x\) in the intersection of the domains of \(f\) and \(g\)).

Answer:

The statement \(f(x)+g(x) = g(x)+f(x)\) is true for all functions \(f\) and \(g\) for which the sum \(f(x)+g(x)\) (and thus \(g(x)+f(x)\)) is defined. This is because the addition of the output values of the functions (which are real numbers, in the typical case of real - valued functions) follows the commutative property of addition, where for any two numbers \(a\) and \(b\), \(a + b=b + a\). So, for each \(x\) in the common domain of \(f\) and \(g\), \(f(x)\) and \(g(x)\) are numbers (or elements of a commutative ring, in a more general algebraic setting), and their sum is commutative.