Transcript The modulus function
The modulus function
The modulus of x, written
x
and is defined by
x
= x if x
0
x
= -x if x < 0.
Sketch of y =
x
Sketch the curve y = f(x), using a dashed line for points below the x-axis Reflect any part of the curve below the x-axis in the x-axis.
y =
x
y = - x y y =
x
y = x x
Sketch y =
2x - 1
The graph of y =
2
x
- 1
has the right-hand branch with equation y = 2x – 1.
The left-hand branch is the negative of 2x – 1, i.e. y = -2x + 1.
y = -2x +1 y y = 2x - 1 x
Sketch y =
(x-1)(x-2)
3
y
2 1 0 0 -1 1 2 3
x
Sketch y =
cos x
y =
cos x
1.5
0.5
-0.5
0 -1.5
90 180 270 360
x
Equations involving modulus
Solve the equation
2x - 1
= 3
y = 3 Left-hand branch: -2x + 1 = 3
x = - 1 Right-hand branch: 2x - 1 = 3
x = 2 Solution: x = -1 or x = 2
Examples
Solve the equation
2x - 1
=
x - 2
Critical values x < ½ ½
x
2 x > 2
2
x
- 1
x
- 2
-(2x – 1) 2x – 1 2x - 1 -(x – 2) -(x – 2) x - 2 x < ½ : -2x + 1 = -x + 2
½
x
x = - 1 2 : 2x - 1 = -x + 2
x = 1 x > 2 : 2x - 1 = x - 2
x = -1 Not consistent.
Solution: x = -1 or x = 1 Other methods: Square both sides or graphical method.
Examples
Solve
x
=
2x + 1
Square both sides x 2 = (2x + 1) 2 x 2 = 4x 2 + 4x + 1 3x 2 + 4x + 1= 0 (3x + 1)(x + 1) = 0
x = -1/3 or x = - 1 Solve
x-3
=
3x + 1
Square both sides (x – 3) 2 = (3x + 1) 2 x 2 - 6x+ 9 = 9x 2 + 6x + 1 8x 2 + 12x – 8 = 0 2x 2 + 3x – 2 = 0 (2x -1)(x + 2) = 0 x = - 2 or x = ½
Inequalities involving modulus
x
<
a
-a < x < a
x
>
a
x < -a or x > a Example
x
<
2
-2 < x < 2 -2 2
x
>
2
x < -2 or x > 2 -2 2
Examples
Solve the inequality
2x - 1
< 3
y = 3
2x - 1
< 3 - 3 < 2
x
– 1 < 3
-2 < 2x < 4
-1 < x < 2
Examples
Solve the inequality
2x - 3
5
y = 5
2x - 5
5 2
x
– 3 -5 or 2
x
– 3 5
x
-1 or x
4
Examples
Solve the inequality
x + 2
<
3x + 1
Square both sides: (x + 2) 2 < (3x + 1) 2
x 2 + 4x + 4 < 9x 2 + 6x + 1 8x 2 + 2x - 3 > 0 (4x + 3)(2x – 1) > 0
x < - ¾ or x > ½
4 6
y
2 -4 -3 -2 -1 0 0 1 2 3
x
Examples
Solve the inequality
x + 2
> 2
x
+ 1
Critical values x < -2 x
- 2
x + 2
-x - 2 x + 2
2
x
+ 1
x + 1 2x +1 when x < - 2: -x – 2 > 2x + 1
when x
- 2: x + 2 > 2x + 1
x < - 1 x < 1 Solution x < 1
-4 -3 -2 4 6
y
2 -1 0 0 -2 -4 1 2 3
x
Example
A graph has equation y = 2x +
x + 2
. Express y as a linear functions of x ( y = mx + c).
When x < -2 : y = 2x – x – 2 = x – 2
y = x - 2
When x
-2 : y = 2x + x + 2 = 3x + 2
y = 3x + 2