Section 1.1 ● Real numbers Set­Builder notation Interval notation Functions­ a function is the set of all possible points y that are mapped to a single point x. If when x=5 y=4,5 then it is not a function because when graphed it will not pass the vertical line test. x is the independent variable because it represents values in the domain y is the dependent variable because it represents values in the range Implied domain= all of the possible real numbers on the domain that keep the equation real piecewise­defined function= a function defined by multiple equations Section 1.2 Zero­ the x­intercepts Roots­ the solution to the equation when x=0 Line Symmetry­ when a graph is mirror around the vertex Point Symmetry­ When a graph can be rotated 180 degrees about the axis and still be the same Even functions­ Functions symmetric about the y­axis Odd functions­ Functions that are symmetric about the x­axis Section 1.3 Topics: Continuity, end behavior, Formulas: Terms: Discontinuity Functions: these are functions that are not continuous but there are different types of discontinuities. Infinite Discontinuity­ When the value at x=C increases or decreases indefinitely as x approaches c from the left and right. Jump Discontinuity­ A function is this discontinuity at x=c if the limits of the function as z approaches c from the left and right exist but have two distinct values Removable Discontinuity­ Section 1.4 Section 1.5 ➔ Parent Function: the original function that is transformed to create other function of the same family. ➔ Constant Function: has the form f(x)= c ➔ Zero Function: occurs when the c value of a constant function is equal to zero ➔ Identity function: f(x)=x passes through all points (a,a)0 ➔ Quadratic Function: f(x)= x2 is a parent function. ➔ Cubic Function: f(x)=x3 is the parent function. ➔ Square Root Function: f(x)= √x ➔ Reciprocal Function: f(x)=1/x ➔ Absolute Value Function: f(x)=lxl ➔ Greatest Integer Function: type of step function f(x)=[x] the greatest less than or equal to x. ➔ Transformations: the changing of a parent graph that may affect the appearance of the graph but is derived from the parent graphs formula. ➔ Translations: the shifting of a parent graph. Vertical translations shift the graph up or down, while horizontal transformations shift the graph left or right. ➔ Reflections: Mirror image of the graph over a certain axis. ➔ Dilation: The expansion or compression of a graph vertically or horizontally. f(x) = a • (x − h) + k a: control dilation of the graph h: horizontal shift k: vertical shift Sign on a determines whether the graph is reflected over the x­axis. Sign on x determines whether the graph is reflected over the y­axis. 1­6: Function Operations and Composition of Functions In function operations, they give you a formula (shown above) for two functions and tell you to find the sum, product, difference, or quotient for a new, combined function. Function operation is simply adding, multiplying, subtracting, or dividing two formulas. Ex: Given f(x) = x + 3 and g(x) = ­x2 + 5, find (f + g)(x). (f + g)(x) = ((x + 3) + (­x2 + 5)) = (x + 3 + ­x2 + 5) = (­x2 + x + 8) Answer: ­x2 + x + 8 If asked to find (f + g)(x), (f – g)(x), (f×g)(x), and (f / g)(x) with x being a specific value (such as x = 2), simply find the value of the function at given value x into the equations f(x) and g(x) and plug those answers into (f + g)(x), (f – g)(x), (f×g)(x), or (f / g)(x). Ex: Given f(x) = x + 3 and g(x) = ­x2 + 5, find (f×g)(2) f(2) = 5 g(2) = 1 (f×g)(2) = f(2) x g(2) = 5 x 1 = 5 Answer: 5 In a composition, we are trying to find the formula that result from plugging the formula f(x) and g(x). Ex: Given f(x) = x + 3 and g(x) = –x2 + 5 find (f o g)(x) ( f o g)(x) = f (g(x)) = f (–x2 + 5) = ((–x2 + 5) + 3) = (–x2 + 5 +3) = –x2 + 8 Answer: –x2 + 8 Practice Problems: 1. Given f(x) = x2 + x + 4 and g(x) = x ­ 8, find (f + g)(x), (f – g)(x), (f×g)(x), and (f / g)(x). 2 1 2. Given g(x) = –5x + 3x ­ 16 and f(x) = 4x +9 , find (g o f)(x). Answers: 1. (f + g)(x) (f + g)(x) = (x2 + x + 4) + (x ­ 8) = x2 + x + 4 + x ­ 8 = x2 + 2x ­ 4 Answer: (f + g)(x) = x2 + 2x ­ 4 (f ­ g)(x) (f ­ g)(x) = (x2 + x + 4) ­ (x ­ 8) = x2 + x + 4 ­ x + 8 = x2 + 12 Answer: (f ­ g)(x) = x2 + 12 (f × g)(x) (f×g)(x) = (x2 + x + 4)(x ­ 8) = x3 ­ 8x2 + x2 ­ 8x + 4x ­ 32 = x3 ­ 7x2 ­ 4x ­ 32 Answer: (f × g)(x) = x3 ­ 7x2 ­ 4x ­ 32 (f / g)(x) (f / g)(x) = (x2 + x + 4)/(x ­ 8) = Answer: (f / g)(x) = 2. (f o g)(x) = f(g(x)) = f(–5x2 + 3x ­ 16) 1 = 4(–5x2 + 3x − 16) + 9 1 = −20x2 + 12x − 64 + 9 1 = −20x2 + 12x − 55 1 Answer: (f o g)(x) = −20x2 + 12x − 55 Section 1.7 Inverse relationships: Any function that is flipped over the y=x line Inverse functions These functions are inverse to each other The (X)’s and (Y)’s are switched Horizontal Line Test (If the horizontal line hits more than one point than it is not a function) Functions that pass the Horizontal line test are said to be called “One to one” Finding an Inverse Functions: When finding the inverse of an equation, you switch the x’s and y’s and then solve for Y Practice Problems for 1.7: 1) Find the inverse of the equation y=2(x)5 − 3 Answer: y−1 = √5 (x + 3)/2 2) Graph the equation: x2 + 3x + 9 = 0 , then tell me if it passes the horizontal line test. Answer: This is a quadratic function and it does not pass the horizontal line test..