- WE'RE GIVEN F OF X EQUALS
THE SQUARE ROOT OF THE QUANTITY (2X - 1) - 3.
WE WANT TO DETERMINE THE DOMAIN AND RANGE OF THE GIVEN FUNCTION
AND THEN FIND THE INVERSE FUNCTION.
BECAUSE OUR FUNCTION CONTAINS A SQUARE ROOT
IN ORDER FOR THE FUNCTION VALUE TO BE REAL
THE NUMBER UNDERNEATH THE SQUARE ROOT OR THE RADICAND
WHICH IN THIS CASE 2X - 1 CAN'T BE NEGATIVE
WHICH MEANS 2X - 1 MUST BE GREATER THAN OR EQUAL TO ZERO.
SINCE 2X - 1 MUST BE GREATER THAN OR EQUAL TO ZERO
THIS RESTRICTION WILL HELP US FIND OUR DOMAIN.
WE JUST NEED TO SOLVE THIS FOR X,
SO WE'LL ADD 1 TO BOTH SIDES, DIVIDE BY 2,
SO X HAS TO BE GREATER THAN OR EQUAL TO 1/2
WHICH WOULD BE DOMAIN OF F OF X.
IF WE WANTED TO USE INTERVAL NOTATION
THIS WOULD BE THE INTERVAL FROM 1/2 TO INFINITY CLOSED ON 1/2,
MEANING IT INCLUDES 1/2.
TO FIND THE RANGE OF F OF X
WE WANT TO DETERMINE ALL THE POSSIBLE Y VALUES
OF THIS FUNCTION.
WELL THE VALUE OF THIS SQUARE ROOT IS ALWAYS GOING TO BE
GREATER THAN OR EQUAL TO ZERO
SO THE SMALLEST VALUE THIS COULD BE WOULD BE 0 - 3
WHICH WOULD BE -3
AND ALL OTHER FUNCTION VALUES WOULD BE LARGER THAN -3.
SO THE RANGE IS Y IS GREATER THAN OR EQUAL TO -3
OR USING INTERVAL NOTATION
WE WOULD HAVE THE INTERVAL FROM -3 TO INFINITY.
LET'S GO AHEAD AND VERIFY THIS
BY GRAPHING THE ORIGINAL FUNCTION.
AGAIN, HERE'S THE GIVEN FUNCTION,
IF WE WERE TO PROJECT THIS ON TO THE X AXIS
NOTICE HOW THE X VALUES WOULD BE FROM -1/2 TO INFINITY
AND IF WE PROJECTED THIS ONTO THE Y AXIS
NOTICE HOW THE VALUES WOULD BE FROM -3 TO POSITIVE INFINITY.
SO WE HAVE THE DOMAIN AND RANGE CORRECT,
NOW LET'S GO AHEAD AND FIND THE INVERSE FUNCTION.
REMEMBER INVERSE FUNCTIONS UNDO EACH OTHER,
MEANING IF FUNCTION F HAS AN INPUT OF X AND AN OUTPUT OF Y
THIS Y BECOMES THE INPUT INTO THE INVERSE FUNCTION
WHICH RETURNS THE ORIGINAL VALUE OF X.
SO THE PROCESS FOR FINDING THE INVERSE FUNCTION
IS TO WRITE THIS IN TERMS OF X AND Y
AND THEN INTERCHANGE THE X AND Y VARIABLES.
SO WE CAN WRITE THE GIVEN FUNCTION AS
Y = THE SQUARE ROOT OF THE QUANTITY (2X - 1) - 3
WHICH MEANS THE INVERSE FUNCTION WOULD BE THE EQUATION X =
THE SQUARE ROOT OF (2Y - 1) - 3.
AND NOW WE JUST NEED TO SOLVE THIS FOR Y
AND REPLACE Y WITH INVERSE FUNCTION NOTATION.
SO THE FIRST STEP, WE'LL ADD 3 TO BOTH SIDES OF THE EQUATION.
SO WE'LL HAVE X + 3 = THE SQUARE ROOT OF THE QUANTITY 2Y - 1.
NOW TO UNDO THE SQUARE ROOT
WE'LL SQUARE BOTH SIDES OF THE EQUATION.
LET'S GO AHEAD AND LEAVE THIS AS THE QUANTITY X + 3 SQUARED.
ON THE RIGHT SIDE, SQUARING THE SQUARE ROOT
LEAVES US WITH THE RADICAND OF 2Y - 1.
NEXT STEP, WE'LL ADD 1 TO BOTH SIDES OF THE EQUATION.
IT'LL GIVE US THE QUANTITY (X + 3 SQUARED) + 1 = 2Y.
THE LAST STEP, INSTEAD OF DIVIDING BY 2
LET'S MULTIPLY BOTH SIDES BY 1/2.
SO ON THE LEFT SIDE WE'D HAVE
1/2 x THE QUANTITY X + 3 SQUARED + 1 EQUALS--
ON THE RIGHT SIDE, WE JUST HAVE Y.
NOW WE COULD MULTIPLY THIS OUT AND TRY TO SIMPLIFY THIS
BUT I'M GOING TO GO AHEAD AND LEAVE IT IN THIS FORM
BUT WE SHOULD REPLACE Y WITH INVERSE FUNCTION NOTATION.
SO F INVERSE OF X IS EQUAL TO
1/2 x THE QUANTITY X + 3 SQUARED + 1.
NOW IT'S IMPORTANT TO REMEMBER
THE DOMAIN OF THIS INVERSE FUNCTION IS RESTRICTED.
IT'S GOING TO BE THE RANGE OF THE ORIGINAL FUNCTION.
SO LET'S GO AHEAD AND LIST THAT.
THE DOMAIN OF THIS FUNCTION IS GOING TO BE
WHEN X IS GREATER THAN OR EQUAL TO -3
OR THE INTERVAL FROM -3 TO INFINITY.
AGAIN REMEMBER THE OUTPUT OF THE ORIGINAL FUNCTION
BECOMES THE INPUT INTO THE INVERSE FUNCTION.
SO THE RANGE OF F IS THE DOMAIN OF F INVERSE
AND THE RANGE OF THE INVERSE FUNCTION
WILL BE THE DOMAIN OF THE ORIGINAL FUNCTION.
SO OUR DOMAIN IS GOING TO BE Y GREATER THAN OR EQUAL TO 1/2.
WHICH USING INTERVAL NOTATION WOULD BE FROM 1/2 CLOSED ON 1/2
TO POSITIVE INFINITY.
SO HERE'S THE DOMAIN OF THE ORIGINAL FUNCTION,
HERE'S OUR INVERSE FUNCTION
AND THE DOMAIN AND RANGE OF THE INVERSE FUNCTION.
THE LAST STEP, I ALWAYS LIKE TO VERIFY THIS GRAPHICALLY.
REMEMBER IF WE GRAPH THE FUNCTION
AND IT'S INVERSE FUNCTION ON THE SAME COORDINATE PLAN
THE TWO FUNCTIONS SHOULD BE SYMMETRICAL
ACROSS THE LINE Y = X.
HERE'S THE GRAPH OF THE ORIGINAL FUNCTION,
OUR SQUARE ROOT FUNCTION
AND HERE'S THE GRAPH OF THE OUR QUADRATIC FUNCTION
OR THE INVERSE FUNCTION
AND NOTICE HOW THE DOMAIN OF THE INVERSE FUNCTION
HAS BEEN RESTRICTED DUE TO THE RANGE OF THE ORIGINAL FUNCTION.
AND HERE'S THE LINE Y = X
AND NOTICE HOW THE TWO GRAPHS ARE SYMMETRICAL
ACROSS THIS LINE.
OKAY, I HOPE YOU FOUND THIS HELPFUL.