- WE WANT TO FIND THE DOMAIN OF THE SQUARE ROOT FUNCTION F
OF X = THE SQUARE ROOT OF THE QUANTITY -X SQUARED + 4X + 5.
THE DOMAIN OF THE FUNCTION
WILL BE THE SET OF ALL POSSIBLE INPUTS
OR ALL POSSIBLE X VALUES IN THIS CASE.
WHEN CONSIDERING THE SQUARE ROOT FUNCTION,
WE NEED TO RECOGNIZE THAT THE RADICAND
OR THIS EXPRESSION UNDERNEATH THE SQUARE ROOT
CANNOT BE NEGATIVE,
OTHERWISE THE FUNCTION VALUE WOULD NOT BE REAL.
SO OUR RESTRICTION IS THIS QUANTITY HERE
MUST BE GREATER THAN OR EQUAL TO ZERO.
SO BY SOLVING THIS QUADRATIC INEQUALITY,
WE'LL BE DETERMINING THE DOMAIN OF OUR FUNCTION.
SO TO DETERMINE WHEN -X SQUARED + 4X + 5
IS GREATER THAN OR = 0,
WE'LL FIRST DETERMINE THE X VALUES
WHERE THIS IS EQUAL TO ZERO.
AND THEN WE'LL USE THOSE VALUES
TO FORM INTERVALS ON THE NUMBER LINE, AND THEN TEST TO SEE
WHICH INTERVALS SATISFY THIS ORIGINAL INEQUALITY.
SO, AGAIN, THE FIRST STEP IS TO DETERMINE WHEN -X SQUARED
+ 4X + 5 = 0.
THIS IS FACTORABLE.
LET'S GO AHEAD AND FACTOR OUR NEGATIVE OR -1
SO WE HAVE AN LEADING COEFFICIENT THAT'S POSITIVE.
THIS WILL CHANGE THE SIGN OF EACH TERM,
SO WE'LL HAVE X SQUARED - 4X - 5 = 0.
SO WE'LL FACTOR INTO TWO BINOMIAL FACTORS.
NOW X AND X, THE FACTORS OF -5 ADDED TO -4 WOULD BE -5 AND 1.
SO THIS IS = 0 WHEN X = 5 AND X = -1.
LOOKING BACK AT THE ORIGINAL INEQUALITY FOR A MOMENT,
WE WANT TO KNOW WHEN THIS EXPRESSION
IS GREATER THAN OR EQUAL TO ZERO.
WELL, THESE THE TWO VALUES WHERE IT'S EQUAL TO ZERO,
SO THESE ARE TWO SOLUTIONS OR TWO VALUES IN OUR DOMAIN.
BUT NOW WE'LL ALSO USE THESE VALUES
TO FORM INTERVALS IN THE NUMBER LINE
TO SEE WHICH INTERVALS SATISFY THIS INEQUALITY.
SO BECAUSE THESE DO SATISFY THE ORIGINAL INEQUALITY,
WE'RE GOING TO MAKE CLOSE POINTS AT -1 AND 5.
IF THIS WAS ONLY GREATER THAN 0,
WE WOULD ACTUALLY MAKE OPEN POINTS ON THESE TWO VALUES.
NOW WE'RE GOING TO PICK TEST VALUES IN EACH INTERVAL
TO DETERMINE WHICH INTERVALS SATISFY THIS INEQUALITY.
FOR EXAMPLE, LET'S TEST X = -2 ON THE LEFT.
LETS TEST 0 IN THE MIDDLE, AND LETS TEST 6 ON THE RIGHT.
IF THESE VALUES SATISFY THIS BLUE INEQUALITY,
THEN THE ENTIRE INTERVAL IS TRUE, AND THEREFORE THE DOMAIN.
IF THE TEST VALUE IS FALSE,
THEN THE ENTIRE INTERVAL IS FALSE
AND NOT IN THE DOMAIN OF OUR FUNCTION.
SO WHEN X IS -2, YOU WOULD HAVE -2 SQUARED + 4 x -2 + 5.
AND THE QUESTION IS, IS THIS GREATER THAN OR = 0.
WELL, THIS WOULD BE -4 - 8 + 5 GREATER THAN OR = 0.
THIS IS GOING TO BE -7, SO IT'S NOT GREATER THAN OR = 0,
THEREFORE THIS ENTIRE INTERVAL FALSE.
PUT AN F HERE TO REMIND US.
NOW WE'LL TEST X = 0,
SO WE'D HAVE -0 SQUARED + 4 x 0 + 5 GREATER THAN OR = 0.
THIS WOULD JUST BE 5, WHICH IS GREATER THAN OR = 0.
THEREFORE, THIS ENTIRE INTERVAL IS TRUE.
SO I'LL GO AHEAD AND GRAPH THIS INTERVAL.
THIS INTERVAL IS IN THE DOMAIN OF OUR FUNCTION.
AND THE LAST TEST VALUE IS X = 6.
SO WE'D HAVE -6 SQUARED + 4 x 6 + 5 GREATER THAN OR = 0.
IT'S GOING TO BE -36 + 24 THAT'S -12 + 5 IS -7.
WELL, THIS IS NOT GREATER THAN OR EQUAL TO ZERO,
AND THEREFORE THIS INTERVAL HERE ON THE RIGHT IS FALSE.
SO THE DOMAIN OF OUR FUNCTION USING INTERVAL NOTATION
WOULD BE THE CLOSE INTERVAL FROM -1 TO 5,
MEANING IT INCLUDES THE END POINTS.
BUT WE COULD ALSO SAY THAT X IS GREATER THAN OR = -1
AND LESS THAN OR = 5.
NOW, THIS WAS QUITE A BIT OF WORK,
SO I DO WANT TO SHOW HOW WE COULD'VE DETERMINED
THIS DOMAIN A LOT FASTER GRAPHICALLY.
GOING BACK UP TO OUR RESTRICTION OF THE DOMAIN
OR THIS QUADRATIC INEQUALITY HERE,
WE COULD'VE GRAPHED THE FUNCTION F OF X = -X SQUARED + 4X + 5
AND DETERMINED GRAPHICALLY
WHEN THE FUNCTION WAS GREATER THAN OR = 0.
SO LET'S GO AND TAKE A LOOK AT THAT GRAPH NOW.
SO IN RED WE HAVE THE GRAPH OF F OF X = -X SQUARED + 4X + 5.
TO DETERMINE WHICH X VALUE SATISFY THIS INEQUALITY
WE WANT TO KNOW WHEN THIS FUNCTION
IS GREATER THAN OR EQUAL TO ZERO.
WHICH MEANS ONE OF THE Y VALUES GREATER THAN OR EQUAL TO ZERO,
WHICH OCCURS WHEN THE FUNCTION IS ABOVE THE X AXIS.
NOTICE HOW THIS FUNCTION IS ABOVE THE X AXIS
ON THIS INTERVAL HERE.
AND IF WE PROJECT THIS ON TO THE X AXIS
IT WOULD BE THE CLOSED INTERVAL FROM -1 TO 5.
AGAIN, WE'RE GOING TO INCLUDE THE END POINTS
BECAUSE WE WANT TO KNOW
WHEN THIS IS GREATER THAN OR EQUAL TO ZERO.
AND BECAUSE OF THE EQUAL PART, WE INCLUDE THE END POINTS.
SO THIS GRAPH VERIFIES
THAT WE FOUND THE DOMAIN BY HAND CORRECTLY.
SO THIS IS A NICE WAY TO GRAPHICALLY SOLVE
A QUADRATIC INEQUALITY.
BUT OF COURSE THE OTHER WAY
TO DETERMINE THE DOMAIN OF A SQUARE ROOT FUNCTION
WOULD BE TO JUST GRAPH THE ORIGINAL FUNCTION,
WHICH WE SEE HERE IN BLUE.
AND WE CAN SEE FROM THE GRAPH OF THE ORIGINAL FUNCTION,
THE DOMAIN WOULD BE THE CLOSE INTERVAL FROM -1 TO 5.
OKAY, I HOPE YOU FOUND THIS HELPFUL.