Inverse functions

Subhotosh Khan

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I need help on both of them please, 🙏🏻
Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem
 

JeffM

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These are not inverse functions. They are composite functions.

\(\displaystyle f(b) = c \text { and } g(a) = b \implies f(g(a)) = f(b) = c.\)

\(\displaystyle f(b) = c \text { and } g(c) = d \implies g(f(b)) = d.\)

That is all there is to composite functions.

And this is not a problem in pre-algebra.
 

HallsofIvy

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These are not inverse functions. They are composite functions.

\(\displaystyle f(b) = c \text { and } g(a) = b \implies f(g(a)) = f(b) = c.\)

\(\displaystyle f(b) = c \text { and } g(b) = d \implies g(f(b)) = d.\)

That is all there is to composite functions.

And this is not a problem in pre-algebra.
Well they do turn out to be inverse functions because f(g(x))= g(f(x))= x.
 

JeffM

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@Subhotosh Khan

If you feel that my post is irredeemable, please delete it.

Otherwise, please change second line in LaTeX to

\(\displaystyle f(b) = c \text { and } g(c) = d \implies g(f(b)) = g(c) = d.\) .....................................done

Perhaps (see @HallsofIvy ) my comment that this is a problem in function composition rather than inverse functions is misleading. It turns out that if the domain is restricted to non-negative numbers in this specific problem, f and g are inverses. Of course, if f and g are assumed to be inverses, the problem itself is trivial. It seems to me, however, illegitimate to deduce from a problem statement instructing ”Compute f(g(x)) and g(f(x))“ that f and g are inverses. Of course, f and g may be inverses; that is determined by taking their composition. If, however, you think I am being persnickety or confusing, you should of course delete the post or add this paragraph.
 
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lex

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Yes the function \(\displaystyle f(x)=x^2+7, x \in \mathbb{R}\) doesn't have an inverse function, as it's not 1-1. The composite function \(\displaystyle g(f(x))=|x|, x \in \mathbb{R}\)
E.g. the function \(\displaystyle f(x)=x^2+7, x≥0\) has an inverse function and \(\displaystyle f(g(x))=x, x≥7\) and \(\displaystyle g(f(x))=x, x≥0\)
 
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