Division by zero is not defined. This means that the denominator cannot be zero.
Domain: All real numbers except x=-1 and x=1 Points of Discontinuity: Removable at x=1, and non-removable at x=-1
x-intercepts: None
y-intercept: (0,3)
y=3x-3/(x+ 1)(x- 1)
Recall that division by zero is not defined. Therefore, the rational function is undefined where x+ 1=0 and where x- 1=0.
ccc
x+ 1=0 & and & x- 1=0
⇕ & & ⇕
x= -1 & and & x= 1
This means that neither x= -1 nor x= 1 are included in the domain.
Domain
All real numbers except x= -1 and x= 1
Points of Discontinuity
If a real number a is not in the domain of a function, then the function has a point of discontinuity at x=a. Since the domain is all real numbers except x= -1 and x= 1, we can write the points of discontinuity of our function.
Points of Discontinuity
x= -1 and x= 1
To determine whether the discontinuities are removable or non-removable, we will factor the numerator and cancel out any common factors between the numerator and denominator.
Now that we have simplified the equation, let's consider the points of discontinuity, x= -1 and x= 1.
The factor x- 1 is not in the denominator any more, so x= 1 is a removable point of discontinuity.
The factor x+ 1 is still in the denominator, so x= -1 is a non-removable point of discontinuity.
Intercepts
The intercepts of a function are the points where the graph intersects the axes. Let's calculate the intercepts of the given function, if there are any.
x-intercepts
The x-intercepts of a function are the points where the graph intersects the x-axis. At these points, the value of the y-coordinate is zero. Therefore, to find the x-intercepts, we have to substitute 0 for y in the function rule and solve for x.
We found that if y=0, the value of x is 1. However, x=1 is not included in the domain of the function. Therefore, there are no x-intercepts.
y-intercept
The y-intercept of a function is the point where the graph intersects the y-axis. At this point, the value of the x-coordinate is zero.
Therefore, to find the y-intercept, we have to substitute 0 for x in the function rule and solve for y.
