Let f:Rise defined from R to R . find the solution of f(x) =f^-1 (x)?

Answer 1

# f(x) = x #

We seek a function #f:RR rarr RR# such that solution #f(x)=f^(-1)(x)#

That is we seek a function that is its own inverse. One obvious such function is the trivial solution:

# f(x) = x #

However, a more thorough analysis of the problem is of significant complexity as explored by Ng Wee Leng and Ho Foo Him as published in the Journal of the Association of Teachers of Mathematics.

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Answer 2

Check below.

The points in common between #C_f# and #C_(f^(-1))# if they exist they are not always in the bisector #y=x#. Here is an example of such a function: #f(x)=1-x^2# #color(white)(a)# , #x##in##[0,+oo)#

graph{((y-(1-x^2))sqrtx)=0 [-7.02, 7.03, -5.026, 1.994]}

They are however in the bisector only and only if #f# is #↗# increasing.
If #f# is strictly increasing then #f(x)=f^(-1)(x)# #<=># #f(x)=x#
If #f# is not strictly increasing the common points are found by solving the system of equations
#{(y=f(x)" "),(x=f^(-1)(y)" "):}# #<=># #{(y=f(x)" "),(x=f(y)" "):}# #<=>...#
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Answer 3

#f^(-1)(x)=f(x)# #<=>x=1#

#f(x)=x^3+x-1# #color(white)(aa)#, #x##in##RR#
#f'(x)=3x^2+1>0# #color(white)(aa)# , #AA##x##in##RR#
so #f# is #↗# in #RR#. As a strictly monotone function it is also "#1-1#" and as a one to one function it has an inverse.
We need to solve the equation #f^(-1)(x)=f(x)# #<=>^(f↗)f(x)=x# #<=>#
#x^3+x-1=x# #<=># #x^3-1=0# #<=>#
#(x-1)(x^2+x+1)=0# #<=>^(x^2+x+1>0)#
#x=1#
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Answer 4

The solution to ( f(x) = f^{-1}(x) ) is the set of points where the function f intersects its inverse function f^-1. In other words, the solution consists of all x values such that ( f(x) = f^{-1}(x) ). This can be found by solving the equation for x.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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