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].
