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How do you find the inverse of #3^(2x)?

Answer 1

Caleb Alexander

To find the inverse of (3^{2x}), we first express the function in terms of (y):

[ y = 3^{2x} ]

Then, we interchange the variables (x) and (y):

[ x = 3^{2y} ]

Next, we solve this equation for (y):

[ \log_3(x) = 2y ]

[ y = \frac{1}{2} \log_3(x) ]

So, the inverse function of (3^{2x}) is (f^{-1}(x) = \frac{1}{2} \log_3(x)).

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Answer 2

Chloe Alexander

The step by step explanation and working is given below.

To find the inverse of function please follow the following steps.

Step 1: Swap #x# and # y# Step 2: Solve for # y.#

The final answer would be the inverse function.

Our question #3^(2x)#

#y=3^(2x)#

Step 1: Swap #x# and #y#. #x=3^(2y)#

Step 2: Solve for #y#

#log_3(x) = 2y# Using If #a=b^c# then #log_b(a) = c#

#1/2log_3(x) = y# #y=1/2 log_3(x)# #f^-1(x) =l/2log_3(x)# Answer

If converting logarithm to the exponent form is not clear the following steps might help you understand how it is done.

# log(x) = log(3^2y)# #log(x) = 2ylog(3)# #log(x)/(2log(3)) = y# # 1/2(log(x)/log(3)) = y# using change of base rule. #1/2 log_3(x) = y# The inverse function #f^-1(x) = 1/2 log_3(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.