Domain: All real numbers except 3 Range: All real numbers except 4
Practice makes perfect
To graph the given rational function, we will start by identifying the values of a, b, c, and d.
h(x)=8x+3/2x-6 ⇔ h(x)=8x+ 3/2x+(-6)
We see above that a= 8, b= 3, c= 2, and d=- 6. With this information, we will find the asymptotes and graph the function.
Asymptotes
Let's start by calculating the vertical asymptote. To do so, we need to solve the equation cx+d=0. 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=3. 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 substitute ⟶ & y=8/2
& y=4
The horizontal asymptote is the line y=4.
Graph
Let's make a table of values to find points on the curve. Make sure to include values of x to the left and to the right of the vertical asymptote.
x
8x+3/2x-6
h(x)
- 3
8( - 3)+3/2( - 3)-6
1.75
- 1
8( - 1)+3/2( - 1)-6
0.625
1
8( 1)+3/2( 1)-6
-2.75
5
8( 5)+3/2( 5)-6
10.75
7
8( 7)+3/2( 7)-6
7.375
9
8( 9)+3/2( 9)-6
6.25
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 3. The y-variable takes any value except 4. With this, we can write the domain and range.
Domain:& All real numbers except3
Range:& All real numbers except 4
