Rational Functions

Question 1

Write the original function form and the reciprocal function form

Answer 1

F(x) = mx + b

F(x) = 1/mx + b

Question 2

How do you find the domain

Answer 2

• exclude any value of x that makes denominator zero (-d/c)
  
  • that’s the asymptote

Question 3

Key factors when graphing

Answer 3

• asymptotes
• intercepts
• behaviours bear asymptotes
• intervals

Question 4

Linear rational function form

Answer 4

F(x) = (ax + b)/cx + d

Question 5

How to find the horizontal asymptote

Answer 5

• compare degrees of numerator and denominator
• if degrees are same = a/c
• if numerator less = y = 0
• if numerator is more = no asymptote

Question 6

How to find x intercept

Answer 6

-b/a

Question 7

How to find y intercept

Answer 7

B/d

Question 8

How to find the behaviour as x goes closer towards the asymptote

Answer 8

• find asymptote
• if it a^+ then add 0.001 to the asymptote
• if a^- then subtract 0.001 from the asymptote
• it’ll either go to positive or negative infinity

Question 9

How to find the behaviour as the function goes closer to the y asymptote

Answer 9

• find horizontal asymptote
• set test point to either 1000 or -1000
• answer will be either just above the asymptote or below

Question 10

Discontinuity

Answer 10

• 2 reasons (hole in function)
  
  • vertical asymptote
  • removable discontinuity
    
    • when both numerator and denominator have a common factor that can be cancelled out

Question 11

Oblique asymptote

Answer 11

When degree of numerator is EXACTLY 1 greater than the degree of the denominator (function has no horizontal asymptote)

Question 12

How to find oblique asymptote

Answer 12

• factor
• vertical asymptote
• horizontal asymptote
• synthetic division using the vertical asymptote as the divisor, numerator as the quotient and the oblique asymptote is the answer

Question 13

Rational equation format

Answer 13

[P(x)]/[Q(x)] = [R(x)]/[S(x)]

Question 14

Steps to solve rational equations

Answer 14

• identify domain
  
  • values of x that make denominator zero
  • find common denominator
  • simplify the equation
  • solve for x
  • check
    
    • substitute any solution back into the original to ensure none make denominator zero

Question 15

Rational inequalities format

Answer 15

[P(x)]/[Q(x)] > or < 0

Question 16

Steps to solve rational inequalities

Answer 16

• identify domain
  
  • exclude asymptotes
• find critical points
• set numerator to zero
• set denominator to zero
• solve for x
• test intervals
  
  • use critical points to divide number into intervals
• write solution

Question 17

How to solve rational equations and inequalities

Answer 17

Like normal stuff

Question 18

State the function behaviours:

1/(x-2)

Question 19

State the function behaviours:

1/(x+5)

Question 20

• write the equation for the vertical asymptote
• write the equation for the horizontal asymptote
• determine the y int

F(x) = 5/(1-x)

Question 21

• write the equation for the vertical asymptote
• write the equation for the horizontal asymptote
• determine the y int

F(x) = -1/(x+7)

Question 22

Textbook pg 164 question 3a