SOLUTION:
Case: A logarithmic equation
Given:
[tex]f(x)=log_3(15-5x)\text{ +6}[/tex]
Required: To find
A) The domain
B) The range
C) The vertical line of asymptote
Method:
A) The domain: The domain is obtained on the x-axis. It includes all applicable vales of x for the graph.
To solve for the domain:
We check for the possible values of x where the log function part is greater than 0
[tex]\begin{gathered} (15-5x)>0 \\ 15-5x>0 \\ 15-5x>0 \\ 15>5x \\ \text{Dividing through by 5} \\ 3>x \\ x<3 \end{gathered}[/tex]
B) The range of the function f(x) is given as:
[tex]\begin{gathered} \text{Range:} \\ \lbrack f(x)\colon\text{ f(x)>6}\rbrack \end{gathered}[/tex]
C) The vertical asymptote is at x= 3 i.e where the graph does not go beyond towards the x-axis
Final answer:
A) The domain are all real values less than 3
B) The range is all real numbers greater than 6
C) The vertical asymptote is at x= 3
