How do I find the inverse function of #f(x)=e^(3x)-4 #? And what is it's range ?

Answer 1

Change the x and ys to find the inverse algebraically.

#x = e^(3y) -4#
#x +4 = e^(3y)#
#ln(x+ 4) = ln(e^(3y))#
#ln(x + 4) = 3y#
#y = f^-1(x) = 1/3ln(x +4)#
Since the inverse of the function is the original function reflected over the line #y =x#, the domain of the original function becomes the range of the inverse and vice versa. Since #y = e^(3x) -4# has a domain of all the real numbers, #y = 1/3ln(x+ 4)# will have a range of all the real numbers.

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Answer 2

To find the inverse function of ( f(x) = e^{3x} - 4 ), we first swap ( x ) and ( y ) to get ( x = e^{3y} - 4 ). Then, solve for ( y ) to find the inverse function.

[ x + 4 = e^{3y} ]

[ \ln(x + 4) = 3y ]

[ y = \frac{1}{3} \ln(x + 4) ]

The range of the function ( f(x) = e^{3x} - 4 ) is ( (-4, +\infty) ), because ( e^{3x} ) is always positive, and subtracting 4 shifts the range down by 4 units.

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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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