Review Notes for Rational Functions PDF

RATIONAL FUNCTIONS RATIONAL EQUATION GRAPH OF RATIONAL FUNCTION RATIONAL FUNCTION It is a ratio of two polynomial functions. That is p(x) and q(x) are polynomial functions, then 𝑝(𝑥) 𝑓 𝑥 = 𝑞(𝑥) EXAMPLE OF RATIONAL FUNCTION ...

RATIONAL FUNCTIONS RATIONAL EQUATION GRAPH OF RATIONAL FUNCTION RATIONAL FUNCTION It is a ratio of two polynomial functions. That is p(x) and q(x) are polynomial functions, then 𝑝(𝑥) 𝑓 𝑥 = 𝑞(𝑥) EXAMPLE OF RATIONAL FUNCTION RATIONAL EQUATION STEPS IN SOLVING RATIONAL EQUATION 1.Clear denominators by multiplying each term kunin lamang ang LCD by the LCD 2.Simplify and solve familiar equation. SIMPLIFY 3.Verify if each solution obtained is not an CHECKING excluded value.  RATIONAL EQUATION EXAMPLE: GET THE LCD (𝑥 − 3) (𝑥 − 3) (𝑥 + 2) (𝑥 + 2) (𝑥 − 3)(𝑥 + 2) MULTIPLY THE LCD TO THE BOTH SIDE LCD (𝑥 − 3)(𝑥 + 2) (𝑥 − 3)(𝑥 + 2) (𝑥 + 2) 4 = 𝑥 − 3 (9) SIMPLIFY 4𝑥 + 8 = 9𝑥 − 27 COMBINE LIKE TERMS 4𝑥 − 9𝑥 = −8 − 27 −5𝑥 = −35  −5𝑥 = −35 5𝑥 −35 − = −5 −5 𝑥=7 CHECKING: Make all x=7 4 9 = 7−3 7+2 4 9 = 4 9 RATIONAL EQUATION EXAMPLE: GET THE LCD (𝑥 − 2) (𝑥 − 2) MULTIPLY THE LCD (5) 5 TO THE BOTH SIDE (𝑥 − 2)(5) (𝑥 − 2)(5) (𝑥 − 2) (𝑥 − 2) Isa-isa lang ang pagcancel 5𝑥 + 𝑥 − 2 = 2(5) (𝑥 − 2)(5) 6𝑥 − 2 = 10 Simplify and transposition 6𝑥 = 10 + 2 6𝑥 = 12 Then check. 𝑥=2  GRAPH OF RATIONAL FUNCTIONS domain Vertical Asymptotes Horizontal Asymptotes x-intercept y-intercept graph  DOMAIN VERTICAL ASYMPTOTES HORIZONTAL ASYMPTOTES X - INTERCEPT Y - INTERCEPT DOMAIN always look at the DENOMINATOR. EXAMPLE: 𝑥+3 𝑥+3=0 𝑥 = −3 𝑥𝜀𝑅 such that x ≠ −3 DON’T FORGET ALL THE SYMBOLS VERTICAL ASYMPTOTES same as the domain. THIS IS THE VERTICAL LINE EXAMPLE: OF THE GRAPH. kaya nga siya Vertical 𝑥 = −3 eh. HORIZONTAL ASYMPTOTES EXAMPLE: look for the EXPONENT. ito yung may tatlong condition. 𝑦=0 kapag mataas ang degree/exponent ng ilalim y=0 kapag naman equal ang degree/exponent sa taas at baba y = a/b (yun number na katabi ni x) kapag naman mataas yung degree/exponent ng numerator NO H.A. / Possible for slant X - INTERCEPT EXAMPLE: look for NUMERATOR kapag may x ang numerator, mayroon x-intercept. Equal nyo lang sila sa zero 𝑛𝑜 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 kapag constant lang ang numerator, no x-intercept Y - INTERCEPT EXAMPLE: tanggalin lahat ng may X 1 make sure na ititra ang constant. 𝑦= 3 GRAPH EXAMPLE: kunin ang V.A. at H.A. DOMAIN EXAMPLE: 2 2𝑥 + 𝑥 − 1 = 0 always look at the DENOMINATOR. (2𝑥 − 1)(𝑥 + 1) = 0 i-factor muna 2𝑥 − 1 = 0 (𝑥 + 1) = 0 2𝑥 = 1 𝑥 = −1 DON’T FORGET ALL THE SYMBOLS 1 1 𝑥= 𝑥 = −1 𝑥𝜀𝑅 such that x ≠ , −1 2 2 VERTICAL ASYMPTOTES same as the domain. THIS IS THE VERTICAL LINE EXAMPLE: OF THE GRAPH. 1 kaya nga siya Vertical 𝑥= 𝑥 = −1 eh. 2 HORIZONTAL ASYMPTOTES EXAMPLE: look for the EXPONENT. 1 ito yung may tatlong condition. kapag mataas ang degree/exponent ng ilalim 𝑦= y=0 2 kapag naman equal ang degree/exponent sa taas at baba y = a/b (yun number na katabi ni x) kapag naman mataas yung degree/exponent ng numerator NO H.A. / Possible for slant X - INTERCEPT EXAMPLE: 2 𝑥 −4=0 look for NUMERATOR (𝑥 − 2)(𝑥 + 2) = 0 i-factor muna 𝑥 − 2 = 0 (𝑥 + 2) = 0 kapag may x ang numerator, mayroon x-intercept. Equal nyo lang sila sa zero 𝑥 = 2 (𝑥 = −2 kapag constant lang ang numerator, no x-intercept Y - INTERCEPT EXAMPLE: tanggalin lahat ng may X −4 make sure na ititra ang constant. 𝑦= =4 −1 GRAPH EXAMPLE: kunin ang V.A. at H.A.  Domain 𝑥𝜀𝑅 such that x ≠ −4 Vertical Asymptotes 𝑥 = −4 Horizontal Asymptotes 𝑦 =0+1=1 x-intercept 𝑛𝑜 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 1 5 y-intercept 𝑦 = 1 𝑜𝑟 4 4 LASTLY, REMEMBER THE CODE OF SIR MIKE: DENOMINATOR NUMERATOR DDENX DENOMINATOR EXPONENT remove all X Good luck on your exams! God bless.