Answer:
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
