Section 1.1 Real numbers. Set Builder notation. Interval notation

Transcription

1 Section 1.1 Real numbers Set Builder 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 piecewise defined function= a function defined by multiple equations

2 Section 1.2 Zero the x intercepts 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 y axis Odd functions Functions that are symmetric about the x axis

3 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

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5 Section 1.4

6 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

7 Quadratic Function: f(x)= x 2 is a parent function. Cubic Function: f(x)=x 3 is the parent function.

8 Square Root Function: f(x)= x Reciprocal Function: f(x)=1/x

9 Absolute Value Function: f(x)=lxl Greatest Integer Function: type of step function f(x)= [ x] the greatest less than or equal to x.

10 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 x axis. Sign on x determines whether the graph is reflected over the y axis. 1 6: Function Operations and Composition of Functions In function operations, they give you a formula (shown above) for two functions and tell you to

11 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) = x 2 + 5, find (f + g)(x). (f + g)(x) = ((x + 3) + ( x 2 + 5)) = (x x 2 + 5) = ( x 2 + x + 8) Answer: x 2 + 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) = x 2 + 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) = x find (f og)(x)

12 Practice Problems: ( f og)(x) = f (g(x)) = f ( x 2 + 5) = (( x 2 + 5) + 3) = ( x ) = x Answer: x Given f(x) = x 2 + x + 4 and g(x) = x 8, find (f + g)(x), (f g)(x), (f g)(x), and (f / g)(x). 2. Given g(x) = 5x 2 + 3x 16 and f(x) = 1 4x +9, find (g of)(x). Answers: 1. (f + g)(x) (f + g)(x) = (x 2 + x + 4) + (x 8) = x 2 + x x 8 = x 2 + 2x 4 Answer: (f + g)(x) = x 2 + 2x 4 (f g)(x) (f g)(x) = (x 2 + x + 4) (x 8) = x 2 + x + 4 x + 8 = x Answer: (f g)(x) = x (f g)(x) (f g)(x) = (x 2 + x + 4)(x 8) = x 3 8x 2 + x 2 8x + 4x 32 = x 3 7x 2 4x 32 Answer: (f g)(x) = x 3 7x 2 4x 32 (f / g)(x) (f / g)(x) = (x 2 + x + 4)/(x 8) =

13 Answer: (f / g)(x) = 2. (f og)(x) = f(g(x)) = f( 5x 2 + 3x 16) = 1 4( 5x2 + 3x 16) + 9 = 1 20x2 + 12x = 1 20x2 + 12x 55 Answer: (f og)(x) = 1 20x2 + 12x 55 Section 1.7 Inverse relationships: Any function that is flipped over the y=x line

14 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

15 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: x x + 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.