Ex 2 - Composite function values

- WE WANT TO FIND THE COMPOSITE FUNCTION VALUES F OF G OF 2 AND G OF F OF 2 GIVEN F OF X = X SQUARED + 5 AND G OF X = 6X - 15. SO TO FIND F OF G OF 2, IT'S GOING TO BE EASIER IF WE APPLY THE DEFINITION WHERE F OF G IS EQUAL TO F OF G OF X, SEEN HERE. SO WE CAN REWRITE THIS AS F OF G OF 2. AND THEN FROM HERE, WE CAN THINK OF THE ORDER OF OPERATIONS WHERE WE START WITH THE INNERMOST PARENTHESES. SO IN THIS CASE, OUR FIST STEP IS GOING TO BE TO DETERMINE THE FUNCTION VALUE G OF 2. LET'S GO AHEAD AND DO THAT BELOW. AND SINCE G OF X IS EQUAL TO 6X - 15, G OF 2 IS GOING TO BE EQUAL TO 6 x 2 - 15. SO G OF 2 IS GOING TO BE EQUAL TO, THIS WOULD BE 12 - 15 OR -3. AND SINCE G OF 2 IS EQUAL TO -3, THIS COMPOSITE FUNCTION VALUE IT JUST EQUAL TO F OF -3. AND NOW, WE'LL SUBSTITUTE -3 INTO FUNCTION F WHERE F IS EQUAL TO X SQUARED + 5. SO WE'D HAVE -3 SQUARED + 5. WELL, -3 SQUARED, BE CAREFUL HERE, WE'RE SQUARING -3 WHICH MEANS -3 x -3, SO THIS IS +9 + 5 WHICH IS EQUAL TO 14. SO OUR COMPOSITE FUNCTION F OF G OF 2 IS EQUAL TO 14. SO NOTICE HOW IT TAKES 2 SUBSTITUTIONS IN ORDER TO DETERMINE THIS COMPOSITE FUNCTION VALUE. AND NOW FOR THE SECOND EXAMPLE, NOTICE HOW THE ONLY DIFFERENCE IS THE ORDER OF COMPOSITION HAS CHANGED. AND SINCE G OF F IS EQUAL TO G OF F OF X, WE CAN REWRITE THIS AS G OF F OF 2. NOTICE HOW WHEN USING THIS DIFFERENT NOTATION, THE ORDER OF G AND F STAY THE SAME. AND NOW STARTING WITH THE INNERMOST FUNCTION VALUE, WE'LL DETERMINE F OF 2 FIRST. AND WE'LL GO AHEAD AND DO THIS BELOW. AND SINCE F OF X IS EQUAL TO X SQUARED + 5, F OF 2 IS GOING TO BE EQUAL TO 2 SQUARED + 5. WELL, 2 SQUARED IS 4 + 5 IS EQUAL TO 9, SO F OF 2 IS EQUAL TO 9. SO NOW, WE CAN SUBSTITUTE 9 FOR F OF 2. SO OUR COMPOSITE FUNCTION VALUE IS GOING TO BE EQUAL TO G OF 9. AND NOW SINCE G IS EQUAL TO 6X - 15, G OF 9 WILL BE EQUAL TO 6 x 9 - 15, SO WE HAVE 54 - 15 WHICH IS EQUAL TO 39. SO G OF F OF 2 IS EQUAL TO +39. NOTICE HOW THESE COMPOSITE FUNCTION VALUES ARE DIFFERENT AND THAT'S WHY IT'S VERY IMPORTANT TO PAY ATTENTION TO THE ORDER OF THE COMPOSITION. OKAY, I HOPE YOU FOUND THIS HELPFUL.