Answer:
the inverse of f(x) = 3^(sqrt(x-4)) is f^-1(x) = ((ln x)/ln 3)^2 + 4.
Explanation:
To find the inverse of f(x) = 3^(sqrt(x-4)), we can follow these steps:
Step 1: Replace f(x) with y:
y = 3^(sqrt(x-4))
Step 2: Swap x and y:
x = 3^(sqrt(y-4))
Step 3: Solve for y:
Take the natural logarithm (ln) of both sides to bring down the exponent:
ln x = ln(3^(sqrt(y-4)))
Using the rule that ln(a^b) = b ln(a), we can simplify the right side:
ln x = (sqrt(y-4)) ln 3
Divide both sides by ln 3:
(sqrt(y-4)) = (1/ln 3) ln x
Square both sides:
y - 4 = ((ln x)/ln 3)^2
Add 4 to both sides:
y = ((ln x)/ln 3)^2 + 4
Step 4: Replace y with f^-1(x):
f^-1(x) = ((ln x)/ln 3)^2 + 4
Therefore, the inverse of f(x) = 3^(sqrt(x-4)) is f^-1(x) = ((ln x)/ln 3)^2 + 4.