Are there any asymptotes of the function
y= 3/x -1
yes.
where the denominator is zero.
and where y goes when x gets infinitely large.
Are there any horizontal asymptotes for y=3/x -1
yes. what happens to y when x gets large?
x y
10 3/9
100 3/99
1 billion 3/999,999,999
As x gets larger, y gets smaller
To determine the asymptotes of a function, we need to examine the behavior of the function as x approaches certain values.
In the case of the function y = 3/x - 1, there are two types of asymptotes to consider: vertical asymptotes and horizontal asymptotes.
1. Vertical asymptote:
A vertical asymptote occurs when the function approaches infinity (positive or negative) as x approaches a certain value. To find the vertical asymptote, we set the denominator of the function equal to zero (since division by zero is undefined). In this case, the denominator of the function is x.
Setting x = 0, we find that the function is undefined at this point because it results in division by zero. Therefore, x = 0 is a vertical asymptote for the function y = 3/x - 1.
2. Horizontal asymptote:
A horizontal asymptote occurs when the function approaches a specific value (positive or negative) as x approaches infinity or negative infinity. To find the horizontal asymptote, we look at the behavior of the function as x becomes very large or very small.
As x approaches positive infinity (∞), the term 3/x becomes smaller and smaller, approaching zero. So, as x approaches infinity, the function y = 3/x - 1 approaches -1. Therefore, the horizontal asymptote is y = -1.
In summary, the function y = 3/x - 1 has a vertical asymptote at x = 0 and a horizontal asymptote at y = -1.