Domain: All real numbers except 13 Range: All real numbers except 2
Practice makes perfect
To graph the given rational function, we will start by identifying the values of a, b, c, and d.
g(x)=6x-1/3x-1 ⇔ g(x)=6x+( - 1)/3x+(- 1)
We see above that a = 6, b = - 1, c = 3, and d = - 1. With this information, we will find the asymptotes, graph the function, and state the domain and range.
Asymptotes
Let's start by calculating the vertical asymptote. We need to solve the equation cx+d=0 so that we have the value of x that will cause the fraction to be undefined. Since we already know the values of c and d, we can substitute them in this equation and solve for x.
The vertical asymptote is the line x= 13. The horizontal asymptote is given by the equation y= ac. Since we already know the values of a and c, we will substitute them in this equation.
y=a/c ⇒ y=6/3 =2
The horizontal asymptote is the line y=2.
Graph
Let's make a table of values to find points on the curve. Make sure to include values of x that are on both sides of the vertical asymptote.
x
6x-1/3x-1
g(x)=6x-1/3x-1
- 1/3
6( - 13)-1/3( - 13)-1
1 12
0
6( 0)-1/3( 0)-1
1
1/6
6( 16)-1/3( 16)-1
0
1/2
6( 12)-1/3( 12)-1
4
2/3
6( 23)-1/3( 23)-1
3
1
6( 1)-1/3( 1)-1
2
Let's plot and connect the obtained points. Keep in mind that rational functions have two branches. Do not forget to graph the asymptotes!
Domain and Range
To state the domain and range of the function, we will consider our graph.
We see that the x-variable takes any value except 13. The y-variable takes any value except 2.
Domain:& All real numbers except 13
Range:& All real numbers except 2
