– 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.

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

Thanks for making this video. Really helpful. You just save our whole class!

Thank you for saving my life!