Identify the asymptotes, removable discontinuities, and intercepts for the graph of the function.
d(x)=(x^2-12x+20)/3x
To find the asymptotes, removable discontinuities, and intercepts of the given function d(x) = (x^2 - 12x + 20) / (3x), we can follow these steps:
1. Determine the vertical asymptotes:
To find the vertical asymptotes, set the denominator equal to zero and solve for x. In this case, set 3x = 0 and solve for x:
3x = 0
x = 0
So, x = 0 is the vertical asymptote of the function.
2. Find the horizontal asymptote:
To determine the horizontal asymptote, compare the degrees of the numerator and denominator. The exponent of the highest power of x in the numerator is 2, and the exponent of the highest power of x in the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
3. Identify any removable discontinuities (holes):
To find any removable discontinuities (holes) in the graph, factor the numerator:
x^2 - 12x + 20
The numerator cannot be factored into simpler terms as it does not have any real roots. Therefore, there are no removable discontinuities.
4. Find the x and y-intercepts:
To find the x-intercepts, set the numerator equal to zero and solve for x:
x^2 - 12x + 20 = 0
Using the quadratic formula or factoring, we can find that the roots of the equation are:
x = 2 and x = 10
Therefore, the x-intercepts are 2 and 10.
To find the y-intercept, substitute x = 0 into the function:
d(0) = (0^2 - 12(0) + 20) / (3(0))
d(0) = 20 / 0 (division by zero is undefined)
Since division by zero is undefined, there is no y-intercept.
In summary:
- Vertical asymptote: x = 0
- Horizontal asymptote: None
- Removable discontinuities (holes): None
- X-intercepts: 2 and 10
- Y-intercept: None (undefined)
To identify the asymptotes, removable discontinuities, and intercepts for the graph of the function d(x) = (x^2-12x+20)/3x, we need to analyze its behavior.
1. Asymptotes:
An asymptote is a line that the graph of a function approaches but does not cross. There are two types of asymptotes: vertical asymptotes and horizontal asymptotes.
a) Vertical asymptotes:
A vertical asymptote occurs when the denominator of a rational function becomes zero. In this case, we have a vertical asymptote if the value of x makes the denominator, which is 3x, equal to zero. Therefore, x = 0 is a vertical asymptote.
b) Horizontal asymptote:
To find the horizontal asymptote, we need to compare the degrees of the numerator and the denominator of the function. The degree of the numerator is 2, and the degree of the denominator is 1. Since the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
2. Removable discontinuities:
A removable discontinuity occurs when the numerator and the denominator have a common factor that can be canceled out. We can find if there are any removable discontinuities by factoring the numerator and determining if any factors cancel with the denominator.
The numerator, x^2 - 12x + 20, can be factored as (x - 2)(x - 10). On simplifying, we get d(x) = (x - 2)(x - 10) / 3x. Notice that (x - 2) is a common factor in both the numerator and the denominator. Therefore, x - 2 cancels out, and we are left with d(x) = (x - 10) / 3.
Since there are no common factors remaining, there are no removable discontinuities.
3. Intercepts:
The y-intercept is the value of the function when x is equal to zero. Substituting x = 0 into the function, we get d(0) = (0^2 - 12*0 + 20) / (3*0) = 20/0. Dividing by zero is undefined, which means that there is no y-intercept for this function.
The x-intercept is the value of x when the function is equal to zero. To find the x-intercept, we set d(x) equal to zero and solve for x. So, (x - 10) / 3 = 0. Cross-multiplying, we get x - 10 = 0. Solving for x, we find x = 10. Therefore, the x-intercept is x = 10.
In summary:
- Vertical asymptote: x = 0
- Horizontal asymptote: None
- Removable discontinuities: None
- Y-intercept: None
- X-intercept: x = 10
clearly as x --> 0
d ---> about +7/x which is plus oo for x>0 and minus oo for x<0
as x gets very large
d --->x^2/3x or x/3
as x gets big negative,
d --> x/3 again which is negative
the numerator is zero when
x = 2 or x = 10
so those are x axis intercepts