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I have a function two images $f:X \rightarrow Y$ and $g:Y \rightarrow X$

The task is to proove or disproove the following statement:

If $f$ is invertible and $g=f^{-1}$ then $g \circ f = id_{X}$

My guess is that this is true. Question: Is that correct?

solution so far:

If f is invertible, then f has to be bijective. If f is bijective, then f is left- and rightinverse.

So that means that: $f^{-1} \circ f = id_{X}$ and $f \circ f^{-1}= id_X$

$\equiv f^{-1} \circ f = id_X \equiv g \circ f = id_X$

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    What's $f^{-1}$, according to you? – 2017-02-19
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    inverse of $f$, so it is left- and right inverse. That means that it is bijective – 2017-02-19
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    So your question is "Is it true that a specific combination of words means its own meaning?". Yes, it always does. – 2017-02-19

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The definition of "$f$ is invertible and $g = f^{-1}$" is "$f\circ g = id_Y$ and $g\circ f = id_X$". So the answer to your question is "Yes, by definition".