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WHAT IS A RATIONAL FUNCTION?

• A rational function is a ratio of two polynomial functions where the polynomial in the denominator does not equal zero

RATIONAL FUNCTIONS

• First, always factor if possible.

• Then, divide out common terms.

2 16( )

4

xf x

x

ARE THEY THE SAME??

No!

2 16( )

4

xf x

x

( ) 4f x x

REMOVABLE DISCONTINUITY (HOLE)

• a single point in which the function has no value

---A hole produces a case where the rational function has for particular x values.

---we want to look for cases where you can cancel a factor from the numerator and the denominator of the rational function

WRITE THE EQUATION:

WRITE THE EQUATION:

1. Write an equation for a function that is a horizontal line with a constant height of 3 but with a hole at x = -2.

2. The same as #1 but with holes at x = -2 and x = 5.

3. Write an equation for a graph that looks like the parabola , but has holes at x = 1 and x = 3.

VERTICAL ASYMPTOTES

• Where are the vertical asymptotes?

• How do we know there is a vertical asymptote by looking at the equation?

1( )f x

x 1

( )3

f xx

1( )

( 3)( 3)f x

x x

VERTICAL ASYMPTOTES:

A non-removable discontinuity that produces a vertical line in which the function approaches but never touches

A VA produces a case where the rational function has for particular x values. (where c is a constant)

We want to look for cases where the denominator is equal to zero (after canceling factors that contribute to holes)

WRITE THE EQUATION:

4) Come up with the equation for a function with two vertical asymptotes, at x = -2 and x = 5.

5) The same as #4 but now with vertical asymptotes at x = -2 and x = 5, and a hole at x =3.

DOMAIN

What is going to affect the domain of our functions?

-Removable discontinuities

-Vertical asymptotes

PUT IT ALL TOGETHER!Find all holes, vertical asymptotes, and the domain

for each problem.

𝑓 (𝑥 )=2 𝑥2+13𝑥+15

𝑥2+13 𝑥+40𝑓 (𝑥 )= 2𝑥

2+11𝑥+55 𝑥2−11𝑥+2

𝑓 (𝑥 )=2 𝑥2−4 𝑥

𝑥2+3𝑥𝑓 (𝑥 )= 𝑥2+4 𝑥+3

𝑥3− 𝑥2−2 𝑥

6)

7)

8)

9)

HOMEWORK:

WB p.27-28 #1-7Directions: Find the holes,

vertical asymptotes, and state the domain.