Como sabes si la buzo se mueve en dirección positiva o negativa antes de resolver la ecuación?​

Option A. Domain = (-Infinity, Infinity); Range (0, Infinity)

Solution:

f(x)=2^(3x)

a) This is an exponential function, and we don't have restrictions for the independent variable "x" in the exponent, then the Domain of f(x) is all the real numbers:

Domain f(x) = ( - Infinity, Infinity)

b) To find the range we can find the inverse function f^(-1) (x). The domain of the inverse function is the range of the original function f(x):

y=f(x)

y=2^(3x)

Isolating x: Applying log both sides of the equation:

log y = log 2^(3x)

Applying property of logarithm: log a^b = b log a; with a=2 and b=3x

log y = 3x log 2

Dividing both sides by 3 log 2:

log y / (3 log 2)=3x log 2 / (3 log 2)

log y / (3 log 2)=x

x=log y / (3 log 2)

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

f^(-1) (x) = log x / (3 log 2)

This is a logaritmic function and the argument of the logarithm must be greater than zero, then the Domain of the inverse function is:

x>0→Domain f^(-1) (x) = (0, Infinity)

The domain of the inverse is the range of the original function:

Range f(x) = Domain f^(-1) (x)

Range f(x) = (0, Infinity)

Domain f(x) = ( - Infinity, Infinity)

Range f(x) = ( 0, Infinity)