What is the inverse of the function f (x) = 2^x+6

1) The inverse of the function f (x) = 2^x+6 is: f^ (-1) (x) = log (x-6) / log (2)

2) The inverse of the functio f (x) = 2^ (x+6) is: f^ (-1) (x) = log (x) / log (2) - 6

Solution:

1) f (x) = 2^x+6

y=f (x)

y=2^x+6

Solving for x: Subtracting 6 both sides of the equation:

y-6=2^x+6-6

y-6=2^x

Applying log both sides of the equation:

log (y-6) = log (2^x)

Applying poperty of logarithm: log (a^b) = b log (a); with a=2 and b=x

log (y-6) = x log (2)

Dividing both sides of the equation by log (2)

log (y-6) / log (2) = x log (2) / log (2)

log (y-6) / log (2) = x

x=log (y-6) / log (2)

Changing "x" by "f^ (-1) (x) " and "y" by "x":

f^ (-1) (x) = log (x-6) / log (2)

2) f (x) = 2^ (x+6)

y=f (x)

y=2^ (x+6)

Solving for x: Applying log both sides of the equation:

log (y) = log (2^ (x+6))

Applying poperty of logarithm: log (a^b) = b log (a); with a=2 and b=x+6

log (y) = (x+6) log (2)

Dividing both sides of the equation by log (2)

log (y) / log (2) = (x+6) log (2) / log (2)

log (y) / log (2) = x+6

Subtracting 6 both sides of the equation:

log (y) / log (2) - 6 = x+6-6

log (y) / log (2) - 6 = x

x=log (y) / log (2) - 6

Changing "x" by "f^ (-1) (x) " and "y" by "x":

f^ (-1) (x) = log (x) / log (2) - 6