I know how incredibly stupid this sounds, but bear with me.
Let's take any random $x$, say $3$, and any random $-x$, say $-3$. Let's plug it into $x^2$. They will both give the same result!
I know this conclusion can't be right, that because of the above, $3 = -3$. But how do we logically prove it wrong? I want to know what is logically flawed about the argument above?
I know from the instance above, we can draw 2 conclusions:
- $x$ is really equals to $-x$.
- Just because a function gives the same output for 2 separate numbers doesn't mean the 2 numbers are the same.
Can you please explain why conclusion 2 is the right one to come to?
As an extension, can you please also disprove conclusion 1 above? Why isn't the fact that both provide identical outputs when being plugged into the same function a legitimate reason to say that both inputs are the same? And when is this line of reasoning legitimate; when is it legitimate to say that because both inputs provide the same output, they are the same?
Can you explain all this as simply as possible? I'm still a beginner, and will struggle to understand any rigorous mathematical notation without explanation.