Prerequisite Skills Pg. 2 # 1 7. Properties of Graphs of Functions Pg. 23 # 1 3, 5, Sketching Graphs of Functions Pg.

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1 UNIT FUNCTIONS I Date Lesson Text TOPIC Homework & Video Lesson.0 ().0 Prerequisite Skills Pg. #. (). Functions Pg. # abce,, ace, ace, abc,, 8, 8. (). Absolute Value Pg. # & WS. acegikn 9. (). Properties of Graphs of Functions Pg. #,, 8. (). Sketching Graphs of Functions Pg. #, 9. (). Inverse Relations Pg. #,, 8,,. (). Piecewise Functions Pg. #, 9,. (8). Operations with Functions Pg. #,.8 (9) 9. Composition of Functions Pg. #, ( )doso, 8,, + Famil Tree WS (0) Review for Unit Test Pg. 0 #,,, ab,, ac, 8, 9def, 9, WS. bdfhjlm Pg, # 8, 9 composite functions.0 () UNIT TEST

2 MHF U Lesson. Functions A function is a relation for which there is a unique output for each input. ie: for ever value of the domain there is one and onl one value of the range. Each x value has one value. If ou have a graphical representation of the relation, ou can use the Vertical Line Test (VLT) to determine if it is a function or not. if the graph of the relation never touches a vertical line in more than one place, the relation is a function. Functions can be represented graphicall, numericall, or algebraicall. Graphical Example Numerical Example Algebraic Example Set of Ordered Pairs {(, 0), (, ), (, 0)} Table of Values x x 0 0 Mapping Diagram = x or f( = x 0

3 Ex. Determine whether or not the following are functions, give a reason, and state the domain and range. a) b) = (x ) + c) Ex. Determine an equation for the following functions. a) The input is 0 more than the output. b) The product of the input and output is. Pg. # abce,, ac, ace, abc,, 8,

4 MHF U Lesson. Absolute Value ABSOLUTE VALUE: The distance from zero to a number on a number line The magnitude of a number. ie: For example, speed is the absolute value of velocit. This is because velocit has direction. ie: 0 km/h can mean 0 km/h in a south direction Speed has no direction. It is the magnitude of velocit. A speed of 0 km/h can be in a south or north direction. velocit speed In general, for x : if x 0, x = x if x 0, x = x Ex. Find the value of each of the following. a) b) 8 c) 8 9 Ex. Graph on a number line. a) x b) x Ex. Express each of the following using absolute value notation. a) b)

5 Ex. Sketch the graph of g ( ( x ). x Ex. Solve each of the following absolute value equations. a) x x 9 b) x x b) x x

6 Ex. Solve x x x Pg. # & WS. acegikn

7 MHF U Lesson. PROPERTIES of GRAPHS of FUNCTIONS ODD and EVEN Functions or Smmetr: An EVEN FUNCTION satisfies f( = f( for all its domain. An even function is smmetrical about the axis. ie: For f( = x, f( = ( = x f( = f( and the function is even. x x An ODD FUNCTION satisfies f( = f( for all its domain. An odd function is smmetrical about the origin. ie: For f( = x, f( = ( f( = x = x x f( = f( and the function is odd. x Ex. Determine whether the following are odd, even, or neither. a) g( = x + x b) g( = x c) h( = (x + x ) d) f( = x ( + x x ) e) f( = x f) g( = x x x

8 Ex. Complete the chart below for each of the parent functions listed. Parent Function: The simplest, or base, function in a famil; for example, = x is the parent function for all quadratic functions.

9 Ex. On the grids below, sketch a graph of a function that has the following characteristics. a) VA at x = b) f( increases on the intervals x and x HA at = f( decreases on the interval x f( is increasing on the intervals x and x f( has a slope of zero (0) at x = and x = f( is decreasing on the interval x f(0) = a) x b) x 8 Pg. #,, 8-9 0

10 MHF U Lesson. Sketching Graphs of Functions Transformations Ex. For each of the following, sketch the transformation indicated. a) = f( + b) = f ( x + ) x 8 8 x c) = f (x + ) + d) = f ( x x Ex. Find the image of f( = (, -) if it undergoes the following transformations. a) f ( x ) b) f ( x ) c) f ( x )

11 With respect to the graph of = f( For: = af [k(x d)] + c a: vertical stretch of factor a k: horizontal stretch of factor k d: horizontal translation of d -- if d > 0 right d units -- if d < 0 left d units c: vertical translation of c -- if c > 0 up c units -- if c < 0 down c units For example, for = f [(x + )], the following transformations would be performed on = f(. vertical stretch of factor (ie: multipl values b ) ie: ( ) horizontal stretch of factor (ie: multipl x values b ) ie: (x horizontal translation of --- d = (ie: move points space to left) ie: (x x ) vertical translation of (move points spaces down) ie: ( ) (x, ) ( x, ) Turning Points - a point on a curve where the function changes from increasing to decreasing, or vice versa. For example, the vertex of a parabola. Pg. #, 9

12 MHF U Lesson. INVERSE of a FUNCTION The inverse, f - (, of a function f(, if it exists, is found b writing the equation in the form = f(, interchanging x and, and then solving for OR b reversing the operations of the function. THE INVERSE FUNCTION UNDOES WHAT THE ORIGINAL FUNCTION HAS DONE. The graph of f - (, is a reflection in the line = x of the function f(. Ex. Find f - ( for each of the following. a) f( = (x ) + b) f( = x x Ex. A compan uses the function C( =0.0x + 000, where C is the cost and x is the number of units it produces, to determine its dail costs. Find the inverse of the function and determine how man units are produced when the cost is $

13 Ex. Graph the inverse of each of the following and state, with reason, whether or not the inverse is a function. a) b) x x c) d) x x Pg. #,, 8,,

14 MHF U Lesson. Piecewise Functions Piecewise Function a function defined b two or more rules on two or more intervals. The graph is made up of two or more pieces of different functions x, x 0 ie: f (x ) x, x 0 ( x ), x 0 Ex. Sketch the graph of each of the following. a) f (x ) x x 0 x x 0 x x x b) g( x 0 (x ) x 0 x A piecewise function is said to be continuous if there are no holes or breaks in the graph of the function. Ex. Determine if the following function is continuous. If not, state where it is discontinuous. ( x ), x f (x ) x, x x x, x Ex. Determine the algebraic representation of the following piecewise function.

15 MHF U Lesson. OPERATIONS with FUNCTIONS Addition and Subtraction of Functions When adding or subtracting functions, an x belonging to the domain of f and to the domain of g belongs to the domain of f g. In other words, the domain of f g is the intersection of the domain of f and the domain of g. values that occur in the domains of both f and g. Function f g f + g f g f x g Domain D f D g D f D g D f D g D f D g Ex. If: f {(,),(,),(,),(, ),(0, )} g {(, ), (0,), (,), (,)}, find a) f g b) f g c) g f d) g f The technique of adding/subtracting ordinates ( co-ordinates) is a useful technique for obtaining the graph of functions that ma not easil be obtained otherwise. You onl add the ordinates of equal x co ordinates. f ( x f ( x. g( Graph. g( x Graph h( f ( g( f (, g(, h( h( f ( g( f (, g(, h( x x

16 Ex. Find the graph of the function f (x ) x x, x x 8 9 Ex. Find the graph of the function f ( x x x 8 9 Pg. #, Read over pgs & of lesson.8

17 MHF U Lesson.8 Composition of Functions A composite function is a function that depends on another function. It is formed when one function is substituted into another function. f [ g( ] is the composite function that is formed when g ( is substituted for x in f (. f [ g( ] is read as f at g at x or f of g of x NOTE: The order of the functions is important. As read from left to right, the second function is substituted into the first function. f g ( or f g( are alternative notations that mean the same as f [ g( ]. Domain of a Composite Function At times, the domain of a composite function can be a bit confusing. Let's examine what happens to values as the "travel" through a composition of functions. Consider the following example: f ( and x x g ( What is the domain of ( f g)(? x In this problem, function g( cannot pick up the value x =, and function f ( cannot pick up the value x = -. The domain of ( f g)( will be the values from the domain of g ( which can "move through" to the end of the composition. This means that the answers created b these values from function g( must be "picked up" b function f (.

18 Let's follow this process algebraicall:. Function g( cannot pick up the value. Consequentl, the composition also cannot pick up the value.. The answers coming out of function ( x g come out in the form x. Since function f ( cannot x pick up -, we must lookout for an values of x that cause since these values create an x answer that cannot progress through the composition (cannot be picked up b function f (.) x. When does? Solve algebraicall. x Since the value of x = will create an answer of - from function g (, and - cannot be picked up b function f (, the value x = must be excluded from the domain of the composition.. The domain of ( f g)( will be all real numbers with the exclusion of and. (Notice that one of the excluded values is, not -. The value x = - makes it through the composition ver nicel because its answer from function g( is which is then picked up b function f (.) Is there an easier wa to find the domain of a composition? If ou are finding the algebraic expression for the composition of two functions, ou can examine our answer to determine an additional restrictions on the domain of the composition. Let's continue with our problem... The algebraic expression for this composition SHOWS that x = would not be an acceptable domain element since it creates a zero denominator problem. Just remember that ou must also specif an restrictions on the domain of the starting function. In this problem, x = is not allowed since it is a restriction on g(. Answer: The domain of the composition is all real numbers with the exclusion of and. DOMAIN = { x R, x,}

19 Ex. Let f ( x and g (. Determine an expression for each composite function, sketch the x function and state its domain and range. a) f [ g( ] x b) g[ f ( ] x c) f f ( x

20 d) g g ( 9 8 x e) g [ g( ] x Ex. Let u( x x and w (. x a) Evaluate u [w()] b) Evaluate w [ u( )]

21 Ex. If f ( x, find an expression for g( if ( f g)( x x. Ex. If f = {(, ), (-, ), (, 9), (-, 0)} and g = {(0, ), (, ), (, ), (9, )}, find: a) ( f g)(9) b) ( g f )() c) ( f g g)() SUMMARY Pg. #, ( )doso, 8,, Plus Famil Tree WS To determine the equation of a composite function, substitute the second function into the first, as read from left to right. So, to determine f [ g( ] substitute g ( into f (. To evaluate a composite function f [ g( ] at a specific value, substitute the value into the equation of the composite function and simplif. OR Evaluate g ( at the specific value and then substitute the result into f (