Answer: 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)
Answer:
Domain f(x) = ( - Infinity, Infinity)
Range f(x) = ( 0, Infinity)