Section+1.2

(Yay! Don’t you love functions?) **Function:** a function from a set D to a set R is a rule that assigns to every element in D a __unique__ element in R. Domain – input values (x), independent variable Range – output values (y), dependent variable **Function Notation:** y = f(x) --- “f of x” f is the //name//. (x) is the //input variable//. f(x) = 3x – 1 f(0) = 3(0) – 1 = -1 It is __not__ a function if 2 different y-values have the same x-value. Ex. x = |y| (not a function) Ex. Y = x2 + 1 (function) **Graphically:** Vertical Line Test – for every vertical line that exists, if the line touches the graph at more then 1 point, than it is not a function. 1) Domain – what values can I put in for x? Ex. f(x) = 3x2 – 2x + 5  D: all real numbers  2) Range – what values can I get out of the function? Ex. f(x) = √x + 2 - have fun trying to find all these symbols! D: [-2, ∞) R: [0, ∞) **Continuity:** graph is continuous at a point if the graph does not come apart at that point. 4 types – //continuous//, //removable discontinuity//, //jump discontinuity//, and //infinite discontinuity// (**refer to notes for graphs**) As x increases, over what x-intervals are the y-values increasing, decreasing, or constant? (**refer to notes for graphs**) 1) Bounded Below – the range does not go to - ∞. Lower Bound has the smallest value of the range. 2) Bounded Above – the range does not go to ∞. Upper Bound has the largest value of the range. 3) Bounded – means it has an upper __and__ lower bound. 1) Local – the max. or min. range value of a function of some tiny interval. 2) Absolute – the max. or min. range value of the entire function. Can be both - if it is an absolute, it is also a local. Ex. f(x) = x4 – 7x2 + 6x  (**refer to notes**)  - Hills and valleys are the extrema.  1) Over the y-axis – called even symmetry because y = xn when n is even has this symmetry. Every point (a, b) has a matching point (-a, b). f(-x) = f(x) --- original function 2) Over the origin – called odd symmetry because y = xn where n is odd has this symmetry (rotate 180° and will look same). Every point (a, b) has a matching point (-a, -b).  f(-x) = -f(x) --- opposite of the original function 3) Over the x-axis Every point (a, b) has a matching point (a, -b). (**refer to notes for graphs**) Ex. Tell whether the function is even, odd, or neither. f(x) = x2 – 3 f(-x) = (-x)2 – 3 = x2 – 3 EVEN! Yes, looks like original. g(x) = x2 – 2x – 2 g(-x) = (-x)2 + - 2(-x) – 2 = x2 + 2x – 2 --- No (even), does not look like the original. -g(x) = -x2 + 2x + 2 --- No (odd), is not the opposite of the original function. NEITHER! pg. 102 #1 – 19 odd pg. 102 #21 – 39 odd pg. 102 #41 – 53 odd and #10 – 20 even
 * Section 1.2 Functions and Their Properties **
 * Function or NOT? **
 * Domain and Range: **
 * Increasing, Decreasing, Constant: **
 * Boundaries: **
 * Local and Absolute Extrema: **
 * Symmetry: **
 * Homework Assignments: **