To find the domain and range of the function [tex]\( g(x) = 3 \log_2 x + 1 \)[/tex], follow these steps:
### Domain
1. Identify the base function: The base function here is [tex]\( \log_2 x \)[/tex].
2. Determine the conditions for the logarithm: The logarithm function [tex]\( \log_2 x \)[/tex] is defined only when the argument [tex]\( x \)[/tex] is positive. This means:
[tex]\[
x > 0
\][/tex]
3. Domain conclusion: Therefore, the domain of [tex]\( g(x) \)[/tex] is [tex]\( (0, \infty) \)[/tex].
### Range
1. Consider the range of the base function: The logarithm function [tex]\( \log_2 x \)[/tex] can take any real number value ([tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex]).
2. Effect of multiplication and addition: Multiplying by 3 and then adding 1 to any real number will still cover all real numbers. Hence:
[tex]\[
3 \log_2 x + 1 \text{ can take any real value}
\][/tex]
3. Range conclusion: Therefore, the range of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
Thus, the correct choices are:
- Domain: [tex]\( (0, \infty) \)[/tex]
- Range: [tex]\( (-\infty, \infty) \)[/tex]
The correct answer is:
[tex]$
\text{Domain: } (0, \infty) \text{ and Range: } (-\infty, \infty)
$[/tex]
