Ex:  Find a Composition of Functions Involving Rational Functions
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Ex: Find a Composition of Functions Involving Rational Functions

September 29, 2019


– GIVEN FUNCTIONS F AND G, WE WANT TO DETERMINE THE
COMPOSITE FUNCTION F OF G OF X, AND ALSO G OF F OF X. FOR THE FIRST STEP, LET’S REWRITE THIS
IN THE FORM OF F OF G OF X. THIS FORM’S GOING TO BE EASIER
TO WORK WITH, AND WE’LL WRITE G OF F OF X
IN THIS FORM HERE. SO WE’LL START WITH
THE INNER FUNCTION. NOTICE THERE’S NO INPUT
INTO FUNCTION G, SO WE’RE GOING TO REPLACE G OF X
WITH 3 DIVIDED BY X – 5. SO WE CAN WRITE THIS AS F OF
3 DIVIDED BY X – 5. AND NOW THIS BECOMES THE INPUT
INTO FUNCTION F. SO WE’LL SUBSTITUTE
THIS QUANTITY FOR THIS X HERE, AND THIS WILL GIVE US
OUR COMPOSITE FUNCTION. WE’LL HAVE 1 DIVIDED BY THE
QUANTITY 3 DIVIDED BY X – 5, AND THEN WE STILL HAVE + 5. WELL, THIS ACTUALLY WORKS OUT
PRETTY WELL BECAUSE NOTICE HOW WE HAVE
– 5 HERE + 5 THAT WOULD BE 0. SO THIS SIMPLIFIES NICELY
TO 1/3/X. REMEMBER THIS FRACTION BAR
MEANS DIVISION, SO IF IT’S HELPFUL, WE CAN WRITE THIS AS
A MULTIPLICATION PROBLEM. THIS WOULD BE 1 X THE RECIPROCAL
OF 3/X, WHICH WOULD BE X/3, WHICH OF COURSE IS JUST
X DIVIDED BY 3. SO OUR COMPOSITE FUNCTION F OF G
OF X IS=TO X DIVIDED BY 3. NOW LETS TAKE A LOOK AT
G OF F OF X. AGAIN, WE’LL START WITH
THE INNER FUNCTION F OF X. AND THERE’S NO INPUT
FOR FUNCTION F, SO WE’RE GOING TO REPLACE F OF X WITH 1 DIVIDED BY THE QUANTITY
X + 5. SO THIS IS=TO G OF 1
DIVIDED BY THE QUANTITY X + 5. AND NOW THIS BECOMES
THE INPUT INTO FUNCTION G, WHICH MEANS WE’LL SUBSTITUTE
THIS QUANTITY HERE FOR THE X IN FUNCTION G. SO THIS IS=TO 3
DIVIDED BY 1/X + 5 AND THEN – 5. NOW, FOR THE NEXT STEP, THIS IS
3 DIVIDED BY THIS FRACTION, WHICH IS THE SAME AS 3 x
THE RECIPROCAL OF THIS FRACTION. SO THIS WOULD BE 3
x THE RECIPROCAL OF 1/X + 5 IS JUST X + 5. AND THEN WE STILL HAVE THIS – 5
HERE, SO NOW WE’LL DISTRIBUTE. THIS WOULD BE 3X + 15 – 5,
WHICH IS=TO 3X + 10. SO G OF F OF X IS=TO 3X + 10. I HOPE YOU FOUND THIS HELPFUL.  

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  1. Thank you so much! I was really confused on how to properly compose rational functions, it was frustrating. I understand it now!

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