Question

(1 mark) 3. 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: Analyze the equation

We start with the equation \( f(x) - g(x)=g(x) - f(x) \). Let's rearrange this equation to see what conditions are needed. Add \( g(x) \) and \( f(x) \) to both sides: \( f(x)-g(x)+g(x)+f(x)=g(x)-f(x)+g(x)+f(x) \). Simplifying both sides, we get \( 2f(x) = 2g(x) \). Then divide both sides by 2, we have \( f(x)=g(x) \).

To check if it's not true for all functions, we can take a counterexample. Let \( f(x)=x \) and \( g(x)=0 \). Then \( f(x)-g(x)=x - 0=x \), and \( g(x)-f(x)=0 - x=-x \). Since \( x
eq -x \) for \( x
eq0 \), the equation does not hold for all functions.

Step2: State the condition

From the algebraic manipulation, we saw that \( f(x)-g(x)=g(x)-f(x) \) implies \( 2f(x)=2g(x) \), so \( f(x) = g(x) \). So the statement is not true for all functions. The condition required for the statement to be true is that \( f(x)=g(x) \) (i.e., the two functions are identical).

Answer:

The statement \( f(x)-g(x)=g(x)-f(x) \) is not true for all functions. The condition required for the statement to be true is that \( f(x) = g(x) \) (the two functions are identical). For example, if \( f(x)=x \) and \( g(x)=0 \), \( f(x)-g(x)=x \) and \( g(x)-f(x)=-x \), which are not equal for \( x
eq0 \), showing it's not true for all functions. The condition for the statement to hold is \( f(x)=g(x) \).