continuous discontinuous infinite discontinuous removable discontinuous jump discontinuous

continuous discontinuous infinite discontinuous removable discontinuous - jump discontinuous removable discontinuous jump continuous Pre-Calculus discontinuous - infinite 9/5/2006

(3 x+4)(x<1)+(x-1)(x>1) jump (x^3+1)(x 0)+ (2)(x=0) removable (3+x 2)(x<-2)+(2 x)(x>-2) (x<1)+(11 -x 2)(x>1) jump Pre-Calculus 9/5/2006

incr: (- , ) decr: [ - 1, 1 ] incr: (- , -1 ], [ 1, ) decr: (- , 0 ] incr: [ 0, ) decr: [ 3, 5 ], incr: [ , 3 ] decr: [ 3, ), incr: ( 0 ] constant: [ 5, ) constant: [ 0, 3) decr: ( - , ) decr: (- , -8 ] incr: [ 8, ) Pre-Calculus decr: (- , 0 ] incr: [ 0, ) decr: ( - , 0 ] incr: [ 0, 3 ) constant: [ 3, ) decr: ( 0, ) incr: ( - , 0 ) decr: ( 2, ) incr: ( - , 2) constant: [ -2, 2 ] decr: ( - , 7 ) decr: ( 7, ) 9/5/2006

unbounded Left branch: bounded above B=5 Right branch: bounded below b=5 unbounded below b=0 bounded above B=0 bounded below b=0 Pre-Calculus bounded below b=1 bounded b= -1, B = 1 bounded below b = -1 bounded above B=0 9/5/2006

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, Pre-Calculus f(-x) = - f(x) 9/5/2006

Odd Even Odd Pre-Calculus Even Neither Even Odd 9/5/2006

horizontally vertically will not cross asymptotes tan and cot x = -1 x=2 y=0 End behavior Limit notation Pre-Calculus 9/5/2006

Vertical: x = - 3 Pre-Calculus Horizontal: y = 0 Vertical: x = 2, -2 Horizontal: y = 0 Vertical: x = 3 9/5/2006

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 9/5/2006

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 9/5/2006

10 Basic Functions Pre-Calculus 9/5/2006

In-class Exercise Section 1. 3 Pre-Calculus • Domain • Range • Continuity • Increasing • Decreasing • Boundedness • Extrema • Symmetry • Asymptotes • End Behavior 9/5/2006

f(x) + g(x) f(x) – g(x) f(x)/g(x), provided g(x) 0 3 x 3 + x 2 + 6 3 x 3 – x 2 + 8 3 x 5 – 3 x 3 + 7 x 2 – 7 Pre-Calculus x 2 – (x + 4) = x 2 – x – 4 9/5/2006

x 2 sin(x) +, –, x, applying them in order the squaring function the sin function composition f○g (f ◦ g)(x) = f(g(x)) 4 x 2 – 12 x + 9 1 2 x 2 – 3 5 x 4 Pre-Calculus 4 x – 9 9/5/2006

Pre-Calculus 9/5/2006

inverse functions horizontal line test original relation Graph is a function (passes vertical line test. Inverse is also a function (passes horizontal line test. ) line test like A is paired with a unique y Graph is a function (passes vertical line test. Inverse is not a function (fails horizontal line test. ) both vertical and horizontal one-to-one function is paired with a unique x f – 1 Pre-Calculus inverse function f – 1 (b) = a, iff f(a) = b 9/5/2006

Pre-Calculus 9/5/2006

inside function outside function x 2 + 1 x 2 Pre-Calculus 9/5/2006

{ ( - , ) } { ( - , - 5) U ( - 5, ) } { ( - , ) } Pre-Calculus f(x) and g(x) are inverses 9/5/2006

Yes passes horizontal line test Yes D: { ( - , 0 ) U ( 0, ) } R: { ( - , 4 ) U ( 4, ) } Pre-Calculus D: { ( - , 4 ) U ( 4, ) } 9/5/2006

add or subtract a constant to the entire function f(x) + c f(x) – c up c units down c units add or subtract a constant to x within the function Pre-Calculus f(x – c) right c units f(x + c) left c units 9/5/2006

reflections negate the entire function y = – f(x) negate x within the function y = f(-x) Pre-Calculus 9/5/2006

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 9/5/2006

y = 1/x 4 y = x, y = x 3, y = 1/x, y = ln (x) A n s w e rs y = ln(x) y = 2 sin(0. 5 x) Stretch by 8 Pre-Calculus y = sqrt(x) Shrink by 1/8 Shrink ½ Stretch by 2 9/5/2006