# Precalculus : Rational Functions

## Example Questions

### Example Question #11 : 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.

### Example Question #12 : 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.

### Example Question #13 : 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.

### Example Question #14 : 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.

### Example Question #15 : Find A Point Of Discontinuity

Find the point of discontinuity for the following function:

There is no 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 #16 : Find A Point Of Discontinuity

Given the function, , where and what is the type of discontinuity, if any?

Explanation:

Before we simplify, set the denominator equal to zero to determine where  is invalid.  The value of the denominator cannot equal to zero.

The value at  is invalid in the domain.

Pull out a greatest common factor for the numerator and the denominator and simplify.

Since the terms  can be cancelled, there will not be any vertical asymptotes.  Even though the rational function simplifies to , there will be instead a hole at  on the graph.

### Example Question #17 : Find A Point Of Discontinuity

Find the point of discontinuity in the function .

Explanation:

When dealing with a rational expression, the point of discontinuity occurs when the denominator would equal 0. In this case, so . Therefore, your point of discontinuity is .

### Example Question #1 : Find Intercepts And Asymptotes

Suppose the function below has an oblique (i.e. slant asymptote) at .

If we are given , what can we say about the relation between  and  and between  and ?

Explanation:

We can only have an oblique asymptote if the degree of the numerator is one more than the degree of the denominator.  This stipulates that  must equal .

The slope of the asymptote is determined by the ratio of the leading terms, which means the ratio of  to  must be 3 to 1.  The actual numbers are not important.

Finally, since the value of  is at least three, we know there is no intercept to our oblique asymptote.

### Example Question #2 : Find Intercepts And Asymptotes

Find the -intercept and asymptote, if possible.

Explanation:

To find the y-intercept of , simply substitute  and solve for .

The y-intercept is 1.

The numerator, , can be simplified by factoring it into two binomials.

There is a removable discontinuity at , but there are no asymptotes at  since the  terms can be canceled.

### Example Question #1 : Find Intercepts And Asymptotes

Find the -intercepts of the rational function

.

Explanation:

The -intercept(s) is/are the root(s) of the numerator of the rational functions.

In this case, the numerator is .