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How do you find the inverse of #f(x)= -log_5 (x-3)#?

Answer 1

Lily Anderson

To find the inverse of ( f(x) = -\log_5(x - 3) ), follow these steps:

Replace ( f(x) ) with ( y ): ( y = -\log_5(x - 3) ).
Swap ( x ) and ( y ): ( x = -\log_5(y - 3) ).
Solve for ( y ).
Rewrite the equation in exponential form.
The resulting expression is the inverse function.

So, the inverse of ( f(x) ) is ( f^{-1}(x) ).

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Answer 2

Elias Ackerman

Remember Inverse of #(x,y)# is given by #(y,x)# we are going to use the same to find the inverse. Step by step procedure is given below.

Inverse of #(x,y)# is #(y,x)#

Our process starts by swapping #x# and #y#

#f(x) = -log_5(x-3)#

Remember #y# and #f(x)# are the same.

#y=-log_5(x-3)#

Step 1: Swap #x# and #y#

#x=-log_5(y-3)#

Multiply both sides by #-1#

#-x=log_5(y-3)#

Step 2: Solve for #y#

This requires you to have some knowledge on converting log to exponent form.

#log_b(a) = k => a=b^k#

#log_5(y-3)=-x#

#y-3=5^-x#

Solving for #y#.

Adding #3# on both sides would do the trick here.

#y=5^-x+3#

This #y# is the inverse function and to be represented as #f^-1(x)#

Our answer #f^-1(x) = 5^-x+3#

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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.