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Question 1157859: What happens if we graph both f and f^{-1} on the same set of axes, using the x-axis for the input to both f and f^{-1} ?
[Suggestion: go to www.desmos.com/calculator and type y=x^3 {-2 < x < 2}, y=x^{1/3} { - 2 < x < 2}, and y = x { - 2 < x < 2}, and describe the relationship between the three curves.] Then post your own example discussing the difficulty of graph both f and f^{-1} on the same set of axes.
Suppose f:R \rightarrow R is a function from the set of real numbers to the same set with f(x)=x+1 . We write f^{2} to represent f \circ f and f^{n+1}=f^n \circ f . Is it true that f^2 \circ f = f \circ f^2 ? Why? Is the set { g:R \rightarrow R l g \circ f=f \circ g } infinite? Why?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! here's a reference on the properties of functions that are inverse of each other.
https://www.analyzemath.com/inversefunction/properties_inverse.html
the graph of inverses are reflective of each other across the line y = x.
this can be shown in the following graph.
the red line is the graph of the original equation of y = x^3.
the blue line is the graph of the inverse equation of y = x^(1/3)
the lin y = -x + 4 is there to allow me to show you that the point (x,y) on the graph of the original equation is opposite and equidistant from the line y = x and the reflective point is (y,x) on the graph of the inverse equation.
specifically, the point (1.379,2.621) on the graph of the original equation is equidistant from the line y = x as the point (2.621,1.379) on the inverse equation.
i'm not sure i understand the last part of your question, so i won't try to answer it.
hopefully, what i have provided is helpful to you.
most of the properties of inverse functions are in the reference.
there is another properties of inverse functions that may be what you are alluding to in the last part of your question.
that is that (fog)x) = x if f and g are inverse functions.
in your original example, that is confiemd as shown below.
(fog)(x) = f(g(x)).
(gof(x) = g(f(x)).
if they are inverses of each other, the f(g(x)) = x and g(f(x)) = x.
specifically.
f(g(x)) = (x^(1/3))^3 = x
g(f(x)) = (x^3)^(1/3) = x
check the reference.
lots of good stuff in there.
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