- WE WANT TO DETERMINE THE VALUE OF C
TO MAKE OUR PIECEWISE DEFINED FUNCTION CONTINUOUS EVERYWHERE.
WE HAVE F OF X = X SQUARED + 3X - 4, IF X
AND WE HAVE F OF X = X - 1 IF X > C.
SO NOTICE HOW OUR PIECEWISE DEFINED FUNCTION
IS MADE UP OF A QUADRATIC FUNCTION AND A LINEAR FUNCTION,
AND IF WE GRAPH THESE FUNCTIONS BY THEMSELVES
WITHOUT RESTRICTIONS ON THE DOMAIN THESE TWO FUNCTIONS
WOULD BE CONTINUOUS EVERYWHERE,
WHICH MEANS WE ONLY HAVE TO BE CONCERNED ABOUT THIS FUNCTION
BEING CONTINUOUS AT X = C.
IN ORDER FOR THIS FUNCTION TO BE CONTINUOUS EVERYWHERE,
WE HAVE TO BE ABLE TO SKETCH THIS PIECEWISE DEFINED FUNCTION
WITHOUT LIFTING UP OUR PENCIL.
TO GET A BETTER UNDERSTANDING
OF HOW WE'RE GOING TO APPROACH THIS PROBLEM
LET'S GO AHEAD AND GRAPH THE QUADRATIC FUNCTION
AND LINEAR FUNCTION ON THE SAME COORDINATE PLANE
WITHOUT ANY RESTRICTIONS ON THE DOMAIN.
SO THE RED PARABOLA IS THE GRAPH OF OUR QUADRATIC FUNCTION,
AND THE BLUE LINE IS THE GRAPH OF OUR LINEAR FUNCTION
AND AGAIN, OUR GOAL IS TO FIND THE VALUES OF C
OR TO DETERMINE THESE INTERVALS HERE
SO THAT WE CAN GRAPH THIS PIECEWISE DEFINED FUNCTION
WITHOUT LIFTING UP OUR PENCIL
BECAUSE IF WE CAN DO THAT THE PIECEWISE DEFINED FUNCTION
IS CONTINUOUS EVERYWHERE.
SO WHAT WE'RE GOING TO DO HERE
IS FOCUS ON THESE TWO INTERSECTION POINTS
OF THE PARABOLA AND THE LINE.
ALSO NOTICE THAT WE HAVE THE QUADRATIC FUNCTION
IF X FUNCTION IF X > TC
WHICH MEANS GRAPHICALLY
THE PARABOLA MUST ALWAYS BE TO THE LEFT OF THE LINE
AND THERE'S ACTUALLY GOING TO BE TWO VALUES OF C
THAT WOULD MAKE THIS PIECEWISE DEFINED FUNCTION
CONTINUOUS EVERYWHERE.
ONE VALUE OF C, WHICH WE'LL CALL C SUB 1 = -3
AND THE OTHER VALUE OF C WHICH WE'LL CALL C SUB 2 = +1.
NOTICE HOW IF C = -3 WE'D HAVE THE PARABOLA
WHEN X WHICH WOULD BE THIS PIECE HERE
AND THEN WE'D HAVE THE LINE WHEN X > -3
WHICH WOULD BE THIS PIECE HERE.
NOTICE HOW WE'D BE ABLE
TO SKETCH THIS ORANGE PIECEWISE DEFINED FUNCTION
WITHOUT LIFTING UP OUR PENCIL
AND THEREFORE IT WOULD BE CONTINUOUS EVERYWHERE.
BUT SIMILARLY, IF C = +1 WE COULD GRAPH THE PARABOLA
WHEN X WHICH SHOULD BE THIS PIECE HERE
AND THEN GRAPH THE LINE WHEN X > 1
WHICH WOULD BE THIS PIECE HERE.
AGAIN, NOTICE HOW THIS FUNCTION
WOULD ALSO BE CONTINUOUS EVERYWHERE
BECAUSE WE CAN SKETCH IT WITHOUT LIFTING UP OUR PENCIL.
SO EVEN THOUGH GRAPHICALLY WE KNOW THAT C = -3 OR +1
LET'S ALSO DETERMINE THESE VALUES ALGEBRAICALLY
AND BECAUSE THESE VALUES OF C ARE THE X VALUES
OF OUR POINTS OF INTERSECTION
WE CAN FIND C BY SETTING X SQUARED (+ 3X - 4) = X - 1.
LET'S GO AHEAD AND DO THAT.
SO WE'D HAVE X SQUARED + (3X - 4) = X - 1.
BECAUSE WE HAVE A QUADRATIC FUNCTION,
WE'LL SET THIS EQUAL TO ZERO AND SEE IF IT FACTORS.
SO WE'LL SUBTRACT X ON BOTH SIDES
AND ALSO ADD 1 ON BOTH SIDES.
NOTICE HOW ON THE RIGHT SIDE WE HAVE 0.
ON THE LEFT, WE HAVE X SQUARE + 2X - 3 = 0.
THIS DOES FACTOR NICELY.
THE FACTORS OF -3 THAT ADD TO +2 ARE +3 AND -1.
SO WE HAVE X = -3 OR X = +1
WHICH MEANS AS WE FOUND GRAPHICALLY C = -3 OR +1
AND LET'S GO AHEAD
AND TAKE A LOOK AT THESE TWO GRAPHS ONE MORE TIME.
IF C = -3 WE WOULD HAVE THE PARABOLA
WHEN X WHICH WOULD BE THIS PIECE HERE,
AND THEN WE'D HAVE THE LINEAR WHEN X > -3
WHICH WOULD BE THIS PIECE HERE,
AND NOTICE OUR FUNCTION IS CONTINUOUS EVERYWHERE
BECAUSE WE CAN SKETCH THE FUNCTION
WITHOUT LIFTING UP OUR PENCIL AND THE SECOND VALUE OF C WAS +1
SO WE'D HAVE THE PARABOLA WHEN X
OR THIS PIECE OF THE GRAPH HERE AND THEN WE'D HAVE THE LINE
WHEN X > 1 WHICH WOULD BE THIS PIECE HERE.
AGAIN, NOTICE HOW THIS GRAPH IS ALSO CONTINUOUS EVERYWHERE.
I HOPE YOU FOUND THIS EXAMPLE HELPFUL.