If g is the inverse of f and if #f(x)=x^3-5x^2+2x-1#, how do you calculate g'(-9) if the domain of f(x) is the set of integers less than 0? Thanks in advance for taking time to help me out. Steve?

Answer 1

Use #d/dx(f^-1(b)) = 1/(f'(f^-1(b))#

We have, in this case #b=-9#. We need #f^-1(-9)#.
That is: we need #x# in the domain of #f# such that #f(x) = -9#.
We must solve #f(x) = -9# for solutions in the domain of #f#.
Solve #x^3-5x^2+2x-1= -9#.

Should you possess knowledge of the rational zeros theorem, apply it. If not, utilize "by inspection"—a fancy term for "guess and check"—to solve the problem.

Try #x=1#, #x=2#, #x=-1# STOP!
#f(-x)=-9# and #-1# is in the domain of #f#, so if the problem is well written that should be the only solution in the domain. (If there are multiple solutions, then #f# is not invertible.)
#f'(x) = 3x^2-10x+2#,
and #f(-1) = 15#.
#d/dx(f^-1(-9)) = 1/(f'(f^-1(-9))) = 1/(f'(-1)) = 1/15 #.
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Answer 2

To find ( g'(-9) ), first, we need to find the inverse function ( g(x) ) of ( f(x) ). Then, we evaluate the derivative of ( g(x) ) at ( x = -9 ).

Given ( f(x) = x^3 - 5x^2 + 2x - 1 ), to find the inverse function:

  1. Let ( y = f(x) ).
  2. Swap ( x ) and ( y ): ( x = y^3 - 5y^2 + 2y - 1 ).
  3. Solve for ( y ).

After finding the inverse function ( g(x) ), differentiate it to find ( g'(x) ), and then evaluate ( g'(-9) ).

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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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