The First Degree Rational Function (fractions) |

**Rational Functions (Fractions)**

**Rational** comes from **ratio** -- ratios are fractions-- division.

**Asymptotes**

The first degree rational function always has **two asymptotes**

one vertical and one horizontal.

The** vertical asymptote** occurs because there is a value of ** x that makes** the

The **horizontal asymptote** is best understood with limits in Calculus.

It defines the **value y approaches as the x-values** head for either end of the axis, that is:

**Note:** asymptotes are **LINES** so we have to write an **equation**.

We cannot write "the vertical asymptote is 4". We must write *x* = 4

As we see, the domain and range would both be ** R** if there were no asymptotes

so we describe them exactly that way. We eliminate the asymptote value like this:

domain: |
range: |

If reality dictates, the domain and range could be limited to only positive values.

.

**From General to Standard Form**

Though I much prefer to work with the general form of the rule, we should know how to change from one form to the other. The fraction tells us how -- we have to divide -- then make a minor modification once we're done.

**Example 1**

Put this rational function rule of correspondence into standard form:

We will divide 6*x* – 15 by 2*x* – 8.

The quotient gives us *k* and the remainder gives us *a -- *after the modification.

Now our only problem is that we don't have *x* – *h* in the denominator, we have a multiple of it.

So, we **factor out 2** from 2*x* – 8, then we divide the 9 in the numerator by 2.

Now, we have standard form:

Since *a* = 9/2 > 0, the curve is always **DECREASING** (rational functions are backwards)

The **vertical asymptote** is *x* = 4 and the **horizontal asymptote** is *y* = 3.

The domain is **range:** * *

The *y*-intercept is 15/8 (set *x* = 0) and the zero is 5/2 (set the ** numerator = 0**)

Let's graph it.

Now one from the general form (*easiest in my opinion*)

**Example 2**

Describe the properties and draw the graph of

** Vertical Asymptote**: set 2

** Horizontal Asymptote**: ratio of the coefficients of

** y-intercept**: set

once we have the intercepts, we know the curve is increasing.

**Finding the Inverse Function**

In this case, we'll first **solve for x**,

**Example 3 **Find the inverse function

- 1. cross multiply:

- 2. remove brackets:

- 3. transpose all

- 4. factor out

- 5. divide both sides by

- 6.

- Notice how we put all the

We would get the same result had we switched the variables first, then solved for

.

**Practice**

1) Find the zero and the *y*-intercept for

2) List the **vertical asymptote**, **horizontal asymptote, ***y***-intercept** and **zero** for:

(remember asymptotes are lines -*–* they have equations!!)

a) | b) | c) | d) |

3) Find the inverse function for

4) John calculates his average hourly profit doing landscape work with the rule

,

where P(*x*) is the average hourly profit and *x* is the number of hours he works.

- a) How many hours must John work to break even? (no profit, no loss)

b) What is his hourly profit if he works 8 hours?

c) What is his maximum hourly profit?

5) a) Graph and describe the properties of .

- b) Solve

**Solutions**

1) *x* = – 5/2, *y* = 5

2)

a) VA: |
b) VA: |
c) VA: |
d) VA: |

3) Find the inverse function for

Switch *x* and *y*'s: , then cross-multiply: *x *(*y* + 9) = 2*y* – 5.

we're trying to find *y*, so we multiply out the bracket and put the *y*-terms together on one side:

*xy* + 9*x* = 2*y* – 5 so we transpose to get

*xy* – 2*y* = – 9*x* – 5.

Now we factor out *y*, and divide both sides by it's coefficient:

*y*(*x* – 2) = – 9*x* – 5, so the inverse function is

4)

a) P(x) = 0, so x = 3make numerator = 0 |
b) P(8) = $12.50 | c) maximum < $25 since the horizontal asymptote is y = 25 |

5)a)

Domain: |
Range: |
f (x) > 0x < – 7/3, or x > –1 |
f (x) < 0–7/3 < x < –1 |

b)

so we're really considering the signs of *y* = (*x* + 5)(*x* + 1).

the zeros are at *x* = – 5 and *x* = – 1 -- and since this parabola opens upward,

the *y*-values are negative (< 0) between the zeros.

The solution is – 5 < *x* < – 1 .

(*all content **© MathRoom Learning Service; 2004 - *).