To find the domain and range of the function [tex]\( y = \log_7(3 + 3x) \)[/tex], follow these steps:
### Domain
The domain of a logarithmic function [tex]\( \log_b(a) \)[/tex] requires that the argument [tex]\( a \)[/tex] must be greater than 0 since the logarithm of a non-positive number is not defined.
For the given function [tex]\( y = \log_7(3 + 3x) \)[/tex], the argument of the logarithm is [tex]\( 3 + 3x \)[/tex]. Therefore, we need:
[tex]\[ 3 + 3x > 0 \][/tex]
To solve for [tex]\( x \)[/tex], follow these steps:
[tex]\[ 3 + 3x > 0 \][/tex]
Subtract 3 from both sides:
[tex]\[ 3x > -3 \][/tex]
Divide by 3:
[tex]\[ x > -1 \][/tex]
Thus, the domain of the function in interval notation is:
[tex]\[ (-1, \infty) \][/tex]
### Range
The range of the logarithmic function [tex]\( y = \log_b(a) \)[/tex] where [tex]\( b > 0 \)[/tex] and [tex]\( b \neq 1 \)[/tex], is all real numbers because the logarithm can output any real number depending on the argument's value within its domain.
For our function [tex]\( y = \log_7(3 + 3x) \)[/tex], as long as [tex]\( 3 + 3x \)[/tex] is within its domain (i.e., [tex]\( > 0 \)[/tex]), [tex]\( y \)[/tex] can take any real value from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex].
Thus, the range of the function in interval notation is:
[tex]\[ (-\infty, \infty) \][/tex]
In summary:
The domain is: [tex]\((-1, \infty)\)[/tex]
The range is: [tex]\((-\infty, \infty)\)[/tex]
