Rational Functions-Real

Rational Functions

1. What is it?

   • Form: f(x) = (ax + b) / (cx + d)
   • f(x)=polynomial/polynomial
     • Eg. f(x)= (2x-9)/77x

2. Domain

   • The most IMPORTANT thing
   • Let it strart as {xєℝ}
     • Look at vertical asymptote(s)
       • Exclude those numbers from domain
       • Domain cannot include these numbers

3. Vertical Asymptote

   • Restrictions created by denominator
     • Denominator cannot =0
       • Undefinied
       • Equation of line is y=(value)
   • Function CAN go through asymptote

4. End Behvaviour Asymptote

   • Horizantal Asymptote
     • Divide the numerator by the denominator
       • Only divide till the denominator can evenly go into the numerator
         • write remainder/denominator
           • ANS= quotient (HA) +/- remiander/denominator (distance between HA and function)
       • If denominator cannot go in then EBA=0

5. Solving Rationals

   • Algebraically
     • Equation
       • Factor out numerator and denominator
         • Find the domain restrictions from demoninator
           • Remember the restrictions!
           • Restrictions are the VA
         • Move everything to one side
           • Make one side =0
             • Find common denominator
               • Simplify numerator
                 • Let numerator =0
                   • Solver for variable
                     • Remember domain restrictions!
                   • 'ignore' the denominator for a while
               • Leave denominator factored
                 • Easier to remember restrictions
                 • Useless expanding
           • NEVER cross multiply
             • Will forget restrictions
     • Inequality
       • Solve like normal equation
         • Remember direction of sign (<,>)
   • Graphically
     • Factor denominator and numerator
       • Find zeros
         • Check if restricted
           • If not-mark on graph
       • Find restrictions
         • Draw VA at those points
       • Create interval table
         • Use zeroes, VA, +/- infinity
           • Locate when - and when +
             • Mark direction for VA
     • Horizantal Asymptote
       • Divide numerator by denominator
         • Mark HA on graph

6. End Behaviour

   • Horizantal
     • Used to describe what happens when x is a very large/small number
     • Find HA
       • Use the remainder
         • Sub in a positive value (large)
           • • See what happens at these values (+/-)
               • If (+)ve then from above
                 • As x-->infinity, f(x)--> HA from above
               • If (-)ve then from below
                 • As x-->infinity, f(x)--> HA from below
         • Sub in negative value (large)
   • Vertical
     • Create interval table with zeroes, VA, from +/- infinity
       • Look at where function is (+)ve and (-)ve
         • Find where VA is (not all will be for VA)
           • If (-)ve then goes down towards -inifity
           • If (+)ve then goes up towards +infinity
           • Comes from both sides of VA (from left/ from right)
             • Left: superscript (-)
             • Right: superscript (+)
     • Used to describe what happens when go towards VA

7. Intercepts

   • X-intercepts (ZEROES)
     • Take out numerator
       • Let numerator =0
         • Solve for x
           • Check if restricted
             • If not, its a zero
   • Y-intercepts
     • Sub in 0 for x
       • Simplify
         • Answer= y-intercept
       • If 0 is not a restriction
         • If restriction then no y-intercept