Connect with knowledgeable individuals and find the best answers at IDNLearn.com. Find in-depth and trustworthy answers to all your questions from our experienced community members.

Question 1 of 5

Select the correct answer.

What are the domain and range of this function?

[tex]\[ g(x) = 3 \log_2 x + 1 \][/tex]

A. domain: [tex]$(1, \infty)$[/tex] and range: [tex]$(-\infty, \infty)$[/tex]

B. domain: [tex]$(0, \infty)$[/tex] and range: [tex]$(-\infty, \infty)$[/tex]

C. domain: [tex]$(-\infty, \infty)$[/tex] and range: [tex]$(1, \infty)$[/tex]

D. domain: [tex]$(-\infty, \infty)$[/tex] and range: [tex]$(0, \infty)$[/tex]


Sagot :

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]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for choosing IDNLearn.com. We’re dedicated to providing clear answers, so visit us again for more solutions.