We want to graph the rational functions y= 1x and y= 1x^2. To do so we will first make a table of values. Then we will use those points to identify the shape of the graphs.
y_1=1/x
y_2=1/x^2
x
1/x
(x,y)
1/x^2
(x,y)
- 4
1/- 4
( - 4, - 0.25)
1/( - 4)^2
( -4, 0.06)
- 2
1/- 2
( - 2, - 0.5)
1/( - 2)^2
( -2, 0.25)
- 1
1/- 1
( - 1, - 1)
1/( - 1)^2
( -1, -1)
- 0.5
1/- 0.5
( - 0.5, -2)
1/( -0.5)^2
( - 0.5, 4)
0
1/0
undefined
1/0^2
undefined
0.5
1/0.5
( 0.5, 2)
1/( 0.5)^2
( 0.5, 4)
1
1/1
( 1, 1)
1/1^2
( 1, 1)
2
1/2
( 1, 0.5)
1/2^2
( 1, 0.25)
4
1/4
( 4, 0.25)
1/4^2
( 4, 0.06)
Let's plot these points and connect them with a smooth curve for each function.
b
We are asked to find the vertical and horizontal asymptotes of the functions that we graphed in Part A. Let's take a look at them, considering the lines that the graphs approach.
As we can see, both of the graphs get closer to the y-axis as the x-values get closer to 0, and they get closer to the x-axis as the x-values get larger in absolute value. Therefore, the vertical asymptote of the graphs is the line x=0 and the horizontal asymptote is y=0.
c
We want to find the range of y= 1x and y= 1x^2. Let's remember that the range of a function is the set of all possible y-values. When we consider the graphs that we drew in Part A and B, we can see that the range of y= 1x is all real numbers except zero, and the range of y= 1x^2 is all real numbers greater than zero.
