What is the inverse of #f(x) = (x+1)/(2x+1)#?

Answer 1
#f(x) = (x+1)/(2x+1) = (2(x+1))/(2(2x+1))#
#=((2x+1)+1)/(2(2x+1))#
#=1/2+1/(4x+2)#
Subtract #1/2# from both sides to get:
#1/(4x+2) = f(x)-1/2#

Take the reciprocal of both sides to get:

#4x+2 = 1/(f(x)-1/2)#
Subtract #2# from both sides to get:
#4x = 1/(f(x)-1/2) - 2#
Divide both sides by #4# to get:
#x=1/(4(f(x)-1/2)) - 2#
#=1/(4f(x)-2) - 2#

So

#f^-1(y) = 1/(4y-2) - 2#
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Answer 2

To find the inverse of ( f(x) = \frac{x+1}{2x+1} ), follow these steps:

  1. Replace ( f(x) ) with ( y ).
  2. Swap ( x ) and ( y ).
  3. Solve the resulting equation for ( y ).
  4. Replace ( y ) with ( f^{-1}(x) ) to express the inverse function.

Starting with ( y = \frac{x+1}{2x+1} ):

[ x = \frac{y+1}{2y+1} ]

[ x(2y + 1) = y + 1 ]

[ 2xy + x = y + 1 ]

[ 2xy - y = 1 - x ]

[ y(2x - 1) = 1 - x ]

[ y = \frac{1 - x}{2x - 1} ]

Therefore, the inverse function of ( f(x) ) is ( f^{-1}(x) = \frac{1 - x}{2x - 1} ).

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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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