Question

What is the inverse of f is f(x) = 3^square root of (x-4)

Answer 1

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.