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Consider the given function.

[tex]\[ f(x) = e^{2x} - 4 \][/tex]

To determine the inverse of the given function, change [tex]\( f(x) \)[/tex] to [tex]\( y \)[/tex], switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex], and solve for [tex]\( y \)[/tex]. The resulting function can be written as:

[tex]\[ f^{-1}(x) = \frac{\ln (x + 4)}{2} \][/tex]



Answer :

GinnyAnswer
Absolutely, let's go through the detailed steps to find the inverse of the given function [tex]\( f(x) = e^{2x} - 4 \)[/tex].

1. Rewrite the function using [tex]\( y \)[/tex] instead of [tex]\( f(x) \)[/tex]:
[tex]\[ y = e^{2x} - 4 \][/tex]

2. Switch [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to find the inverse function:
[tex]\[ x = e^{2y} - 4 \][/tex]

3. Solve for [tex]\( y \)[/tex]:

a. Isolate the exponential term:
[tex]\[ x + 4 = e^{2y} \][/tex]

b. Apply the natural logarithm [tex]\( \ln \)[/tex] to both sides to deal with the exponential:
[tex]\[ \ln(x + 4) = \ln(e^{2y}) \][/tex]

c. Simplify the right side using the property [tex]\( \ln(e^a) = a \)[/tex]:
[tex]\[ \ln(x + 4) = 2y \][/tex]

d. Finally, solve for [tex]\( y \)[/tex] by dividing both sides by 2:
[tex]\[ y = \frac{\ln(x + 4)}{2} \][/tex]

4. Express the inverse function:
[tex]\[ f^{-1}(x) = \frac{\ln(x + 4)}{2} \][/tex]

So, after swapping [tex]\( x \)[/tex] and [tex]\( y \)[/tex] and solving for [tex]\( y \)[/tex], we find that the inverse function of [tex]\( f(x) = e^{2x} - 4 \)[/tex] is [tex]\( f^{-1}(x) = \frac{\ln(x + 4)}{2} \)[/tex].