Question

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

Explanation:

Step1: Analyze the domain of \( f(x) = 4^x - 8 \)

The function \( 4^x \) is an exponential function. For any exponential function of the form \( a^x \) (where \( a>0, a
eq1 \)), the domain is all real numbers because we can raise \( a \) to any real - valued exponent. So, for \( f(x)=4^x - 8 \), there are no restrictions on the input \( x \). The domain is the set of all real numbers, i.e., \( \{x|x\text{ is a real number}\} \).

Step2: Analyze the range of \( f(x) = 4^x - 8 \)

We know that for the exponential function \( y = 4^x \), the range of \( 4^x \) is \( \{y|y > 0\} \) because an exponential function with a positive base (greater than 1 in this case, \( 4>1 \)) will always give positive output values. When we subtract 8 from \( 4^x \) to get \( f(x)=4^x - 8 \), we are shifting the graph of \( y = 4^x \) down by 8 units. So, if \( 4^x>0 \), then \( 4^x - 8>- 8 \). So the range of \( f(x)=4^x - 8 \) is \( \{y|y > - 8\} \).

Answer:

domain: \(\{x\mid x\text{ is a real number}\}\); range: \(\{y\mid y > - 8\}\) (the third option)