# Precalculus : Rational Functions

## Example Questions

### Example Question #1 : Find A Point Of Discontinuity

What are the holes or vertical asymptotes, if any, for the function:

Explanation:

Factorize the numerator for the function:

The removable discontinuity is  since this is a term that can be eliminated from the function.  There are no vertical asymptotes.

Set the removable discontinutity to zero and solve for the location of the hole.

The hole is located at:

### Example Question #2 : Find A Point Of Discontinuity

For the following function,   , find all discontinuities, if possible.

Explanation:

Rewrite the function  in its factored form.

Since the  term can be cancelled, there is a removable discontinuity, or a hole, at .

The remaining denominator of  indicates a vertical asymptote at .

### Example Question #3 : Find A Point Of Discontinuity

If possible, find the type of discontinuity, if any:

Explanation:

By looking at the denominator of , there will be a discontinuity.

Since the denominator cannot be zero, set the denominator not equal to zero and solve the value of .

There is a discontinuity at .

To determine what type of discontinuity, check if there is a common factor in the numerator and denominator of .

Since the common factor is existent, reduce the function.

Since the  term can be cancelled, there is a removable discontinuity, or a hole, at .

### Example Question #4 : Find A Point Of Discontinuity

Find the point of discontinuity for the following function:

There is no point of discontinuity for the function.

Explanation:

Start by factoring the numerator and denominator of the function.

A point of discontinuity occurs when a number  is both a zero of the numerator and denominator.

Since  is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the  value, plug in  into the final simplified equation.

is the point of discontinuity.

### Example Question #5 : Find A Point Of Discontinuity

Find the point of discontinuity for the following function:

There is no point fo discontinuity for this function.

Explanation:

Start by factoring the numerator and denominator of the function.

A point of discontinuity occurs when a number  is both a zero of the numerator and denominator.

Since  is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the  value, plug in  into the final simplified equation.

is the point of discontinuity.

### Example Question #6 : Find A Point Of Discontinuity

Find a point of discontinuity for the following function:

There are no discontinuities for this function.

Explanation:

Start by factoring the numerator and denominator of the function.

A point of discontinuity occurs when a number  is both a zero of the numerator and denominator.

Since  is a zero for both the numerator and denominator, there is a point of discontinuity there. Since the final function is  and  are points of discontinuity.

### Example Question #7 : Find A Point Of Discontinuity

Find a point of discontinuity for the following function:

There are no points of discontinuity for this function.

Explanation:

Start by factoring the numerator and denominator of the function.

A point of discontinuity occurs when a number  is both a zero of the numerator and denominator.

Since  is a zero for both the numerator and denominator, there is a point of discontinuity there. Since the final function is  and  are points of discontinuity.

### Example Question #8 : Find A Point Of Discontinuity

Find a point of discontinuity in the following function:

There is no point of discontinuity for this function.

Explanation:

Start by factoring the numerator and denominator of the function.

A point of discontinuity occurs when a number  is both a zero of the numerator and denominator.

Since  is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the  value, plug in  into the final simplified equation.

is the point of discontinuity.

### Example Question #9 : Find A Point Of Discontinuity

Find the point of discontinuity for the following function:

There is no point of discontinuity.

Explanation:

Start by factoring the numerator and denominator of the function.

A point of discontinuity occurs when a number  is both a zero of the numerator and denominator.

Since  is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the  value, plug in  into the final simplified equation.

is the point of discontinuity.

### Example Question #10 : Find A Point Of Discontinuity

Find the point of discontinuity for the following function:

There is no point of discontinuity for this function.

Explanation:

Start by factoring the numerator and denominator of the function.

A point of discontinuity occurs when a number  is both a zero of the numerator and denominator.

Since  is a zero for both the numerator and denominator, there is a point of discontinuity there. To find the  value, plug in  into the final simplified equation.

is the point of discontinuity.