Question

what are the domain and range of $f(x) = 4^x - 8$?
○ domain: {x | x is a real number}; range: {y | y > -8}
○ domain: {x | x is a real number}; range: {y | y > 8}
○ domain: {x | x > 4}; range: {y | y > -8}
○ domain: {x | x > -4}; range: {y | y > 8}

Explanation:

Step1: Find the domain

For the exponential function $4^x$, there are no restrictions on the input $x$ (any real number can be used as an exponent). So the domain of $f(x)=4^x - 8$ is all real numbers.

Step2: Find the range

The range of $4^x$ is $y > 0$ (since exponential functions with positive base never equal or go below 0). Subtract 8 from this range:
$4^x - 8 > 0 - 8$
Simplify to get $y > -8$.

Answer:

domain: $\{x \mid x \text{ is a real number}\}$; range: $\{y \mid y > -8\}$