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Here is the question I am dealing with:

Disprove the statement: Let $f$ be a real-valued function with domain $(-∞,∞).$ Then $∀x,y {∈} (-∞,∞), \,f(x) = f(y)$ implies that $x = y.$

I am not looking for an answer to this problem, but I'm stuck on what it's asking me to disprove. Specifically, what does it mean for $f(x)=f(y)?$

ryang
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Olivia
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Jul 14 '23 at 03:49
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    $f(x)$ is just the number that $f$ sends $x$ to. So if $f(x) = f(y)$ it just means the function $f$ sent $x$ and $y$ to the same place. – Shou Jul 14 '23 at 04:01
  • Counter example : $~\forall x, ~f(x) = 0.$ – user2661923 Jul 14 '23 at 05:28

3 Answers3

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Disprove the statement: Let $f$ be a real-valued function with domain $(-∞,∞).$ Then $∀x,y {∈} (-∞,∞), \,f(x) = f(y)$ implies that $x = y.$

I am not looking for an answer to this problem, but I'm stuck on what it's asking me to disprove.

You're being asked to disprove

  • for every real-valued function $f$ on $\mathbb R,$ for every $(x,y) {∈} \mathbb R^2,$ $$f(x) = f(y)\implies x = y.$$

So, just construct a single real-valued function on $\mathbb R$ and exhibit a single element of $\mathbb R^2$ that violates the given implication, that is, such that the implication's left side is true but its right side is false. For example, find an $f$ such that $f(2)=f(5).$

ryang
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Suppose that you have a function $f$. This question asks you to determine whether it is true that if this function takes the same $y$-value at two points, is it true that these $y$-values correspond to the same $x$-value given to the function. You should provide an example of a function which fails to agree with this statement.

HINT: Horizontal line test

JayP
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It is asking you to disprove f(x)=f(y) always meaning x=y. You could try doing this by showing that that only applies to a particular type of functions and not all functions. Or you could try finding an example that goes against this statement.

Joseph
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