R R { [ -8, ) } R { [ 0, ) } { [ 4, ) } { [ 0, ) } { (- , 3 ] } { (- , 4 ] U [ 2, ) } { (- 3, ) } { (- , -1) U [ 0, ) } { [ 0, ) } Pre-Calculus

discontinuous - infinite discontinuous - infinite removable continuous discontinuous removable discontinuous jump discontinuous - jump continuous continuous continuous discontinuous - infinite discontinuous - infinite Pre-Calculus

(3x+4)(x<1)+(x-1)(x>1) (3+x2)(x<-2)+(2x)(x>-2) jump (x^3+1)(x0)+ (2)(x=0) removable (3+x2)(x<-2)+(2x)(x>-2) (x<1)+(11-x2)(x>1) jump Pre-Calculus

incr: (- , ) decr: (- , 0 ] incr: [ 0, ) decr: (- , 0 ] incr: [ 0, ) decr: [ - 1, 1 ] incr: (- , -1 ], [ 1, ) decr: [ 3, 5 ], incr: [ , 3 ] constant: [ 5, ) decr: [ 3, ), incr: ( 0 ] constant: [ 0, 3) decr: ( - ,  ) decr: ( 0,  ) incr: ( - , 0 ) decr: (- , -8 ] incr: [ 8,  ) decr: ( 2,  ) incr: ( - , 2) constant: [ -2, 2 ] decr: ( - , 0 ] incr: [ 0, 3 ) constant: [ 3,  ) decr: ( - , 7 ) decr: ( 7,  ) Pre-Calculus

bounded below unbounded bounded below b = 0 b = 1 bounded above Left branch: bounded above B = 5 unbounded bounded above B = 0 bounded b= -1, B = 1 Right branch: bounded below b = 5 bounded below b = 0 bounded below b = -1 bounded below b = 0 bounded above B = 0 Pre-Calculus

The graph looks the same to the y-axis EVEN functions The graph looks the same to the left of the y-axis as it does to the right For all x in the domain of f, f(-x) = f(x) x-axis The graph looks the same above the x-axis as it does below it (x, - y) is on the graph whenever (x, y) is on the graph origin ODD functions The graph looks the same upside Down as it does right side up For all x in the domain of f, f(-x) = - f(x) Pre-Calculus

Odd Even Even Odd Even Neither Even Neither Even Odd Pre-Calculus

horizontally vertically will not cross asymptotes tan and cot x = -1 End behavior Limit notation Pre-Calculus

Horizontal: y = 0 Vertical: x = 2, -2 Horizontal: y = 0 Pre-Calculus

Each x-value has only 1 y-value Yes Each x-value has only 1 y-value { ( - , -1 ) U (-1, 1) U (1,  ) } { ( - , 0) U [ 3,  ) } Infinite discontinuity Decreasing: (- , -1), (-1, 0 ] Increasing: ([ 0, 1), (1,  ) Unbounded Left piece: B = 0, Middle piece b = 3, Right piece B = 0 Local min at (0, 3) Even Horizontal: y = 0, Vertical: x = -1, 1 Pre-Calculus

Each x-value has only 1 y-value Yes Each x-value has only 1 y-value { ( - ,  ) } { [ 0,  ) } continuous Decreasing: (- , 0 ] Increasing: [ 0,  ) Bounded below b = 0 Absolute min = 0 at x= 0 Neither even or odd none { ( - , -3 ] U [ 7,  ) } Pre-Calculus

10 Basic Functions Pre-Calculus

In-class Exercise Section 1.3 Domain Range Continuity Increasing Decreasing Boundedness Extrema Symmetry Asymptotes End Behavior Pre-Calculus

f(x)/g(x), provided g(x)  0 3x3 + x2 + 6 3x3 – x2 + 8 3x5 – 3x3 + 7x2 – 7 x2 – (x + 4) = x2 – x – 4 Pre-Calculus

sin(x) x2 +, –, x,  applying them in order the squaring function the sin function function composition f ○ g (f ◦ g)(x) = f(g(x)) 4x2 – 12x + 9 1 2x2 – 3 5 x4 4x – 9 Pre-Calculus

Pre-Calculus

both vertical and horizontal inverse functions horizontal line test original relation Graph is a function (passes vertical line test. Inverse is also a function (passes horizontal line test.) Graph is a function (passes vertical line test. Inverse is not a function (fails horizontal line test.) both vertical and horizontal line test like A one-to-one function is paired with a unique y is paired with a unique x inverse function f –1 f –1 (b) = a, iff f(a) = b Pre-Calculus

Pre-Calculus

D: { ( - ,  ) } R: { ( - ,  ) } D: { ( - ,  ) } D: { ( - , - 2) U ( -2,  ) } R: { ( - , 1) U (1,  ) } D: { ( - , 1) U (1,  ) } Pre-Calculus

inside function outside function x2 + 1 x2 Pre-Calculus

f(x) and g(x) are inverses { ( - ,  ) } { ( - ,  ) } { ( - ,  ) } { ( - , - 5) U ( - 5, ) } { ( - ,  ) } { ( - ,  ) } f(x) and g(x) are inverses Pre-Calculus

passes horizontal line test Yes passes horizontal line test Yes D: { ( - , 0 ) U ( 0,  ) } R: { ( - , 4 ) U ( 4,  ) } D: { ( - , 4 ) U ( 4,  ) } Pre-Calculus

D: { ( - , - 2 ) U ( - 2, 1 ) U ( 1,  ) } D: { ( - , 2/3 ) U ( 2/3, 1 ) U ( 1,  ) } D: { (- , - 2) U (- 2, 1) U ( 1,  ) } D: { ( - , 0 ) U ( 0,  ) } Pre-Calculus

add or subtract a constant to the entire function f(x) + c up c units f(x) – c down c units add or subtract a constant to x within the function f(x – c) right c units f(x + c) left c units Pre-Calculus

negate the entire function y = – f(x) reflections negate the entire function y = – f(x) negate x within the function y = f(-x) Pre-Calculus

multiply c to the entire function Stretch if c > 1 Shrink if c < 1 multiply c to x within the function Stretch if c > 1 Shrink if c < 1 A reflection combined with a distortion complete any stretches, shrinks or reflections first complete any shifts (translations) Pre-Calculus

Answers y = 1/x 4 y = x, y = x3, y = 1/x, y = ln (x) y = sqrt(x) y = 2sin(0.5x) Stretch by 8 Shrink ½ Shrink by 1/8 Stretch by 2 Pre-Calculus