Prove that if f is an invertible function and g is an inverse of f, then Cg =Df

Question

Prove that if f is an invertible function and g is an inverse of f, then Cg =Df and Cf = Dg It is important to include both f o g = IDg and g o f = IDf in the definition of inverse functions, as Example 45 will show. Let f : R+ rightarrow R be defined by f(x) = x and let g : R rightarrow R+ be defined by g(x) = |x|. Then (g o f)(x) = g(x) = x for all x R+; that is, g o f = IR+. However, (f o g)(x) = |x| for all x R, which is not IR. So, f and g are not inverse functions. We have f is one-to-one but not onto, and g is onto but not one-to-one. Prove that a function is invertible if and only if it is objective.

Explanation / Answer

x belongs to Cg implies g(y)=x for some x, y=ginverse(x) so x belongs t domain of f,

so Cg subset of Df

similarly he other inclusion, so proved 44

suppose not bijective

not one one say

then f(x)=f(y) for x not equal to y

f(x) and f(y) has different functional value under g so not wcell defined

if not onto then g does not take all values of its domain so f must be bijective

if f bijective define g=f inverse in the following way

f(x)=y

degine g(y)=x it is well defined as one one onto f

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