Functions Questions and Answers

3 SECTION 3 1 a c 3 5 3 2 33 5 e A b 2 B 2 2 7 d d30 d5 x d6 f 5s 2 x 125s3 36

1 Onto one to one bijection Consider the function f NU 0 NU 0 where f is defined as x 1 if x is even x 1 f x x 1 if x is odd 1 Is f is one to one If yes prove it otherwise provide a counter example 2 Is f is onto If yes prove it otherwise provide a counter example 3 Is f is bijection If so find f 1 inverse of f

Two teams of scientists explore a cave system inside a mountain From the entrance to the cave Team A hikes 192 ft down to an underground lake Team B hikes 175 ft up to explore an open space Which statement is correct regarding the position of the two teams relative to the cave s entrance O Team A is farther from the entrance than Team B because 192 175 O Team A is farther from the entrance than Team B because 192 175 O Team B is farther from the entrance than Team A because 192 175 O Team B is farther from the entrance than Team A because 192 175

The table gives temperatures in degrees Fahrenheit in a certain town on a given day The sinusoidal functioni F t 8 cos t c 30 models the data where F t is the temperature in F at time t in hours past 12 midnight and c is a constant Which of the following is true about the value of e A B D The value of cis 2 because this accounts for a phase shift of g t 8 cos t 30 that aligns a minimum value of the data set with a minimum value of g The value of cis 2 because this accounts for a vertical shift of g t 8 cos t 30 that aligns a minimum value of the data set with a minimum value of g The value of c is 14 because this accounts for a phase shift of g t 8 cos t 30 that aligns a maximum value of the data set with a maximum value of g The value of cis 14 because this accounts for a vertical shift of g t 8 cos t 30 that aligns a maximum value of the data set with a maximum value of g

ask 3 Analyzing a Determine the zeros algebraically and explain what these mean for your situation b Determine the vertex and explain what this means for your situation

Task 2 Equation of the situation a Determine the equation of the situation in standard form Write the equation using function notation Use a graphing app Like Desmos GeoGebra or Google Sheets create the graph and include it in your submission b Convert the equation to vertex form algebraically using complete the square method Write the equation using function notation c Describe the transformations applied to y x2 to get the resulting equation

Let S and T be two sets defined as S 1 2 3 5 and T 2 4 6 7 8 9 10 Let f be a binary relation from S to T defined by the set of ordered pairs 1 Is fa function f 1 2 2 4 3 4 5 6 2 What is the domain of f codomain of f 3 What is the image of f 4 What is the image of 2 and the image of 3 5 What is the pre image of 2 6 Is this a one to one function

4 Composition of functions For each of the following pairs of functions f and g calculate go f and fog 1 f 1 2 3 4 5 6 and g 2 1 4 3 6 5 2 f x 2x 1 and g x 4x 5 for all x Z

5 Inverse functions Let f be a bijective function f RR defined by f x 4x 5 1 Find the inverse of f 2 Calculate f 1 of and fof 1 What do you notice

B C R 2 3 as MAX es 06 In the figure shown point B and the center of each circle lie on AC The ratio of AB to BC is 4 to 1 If the area of the small circle is 72 what is the area of the shaded region TE

TASK 2 Modelling a function 1 The following function h x 2 x 12 5 has undergone a horizontal transition o the right a vertical transition up by 3 and stretched by the factor of 4 What would be function 3 marks

2 a Write an equation of the form y a x h k with vertex 2 10 that models the flight path for the toy rocket assuming the rocket leaves the ground at 0 0 5 marks Hint to find the value of a use the coordinates of a point where rocket leaves the groung and your knowledge of h and k values of the vertex form b What are the domain and range of the function What do they represent in this situation 3 marks

The graph of the quadratic function f x x 2 x 8 intersects the y axis at 0 c a Find the value of c C 8 The vertex of the function is 3 12 5 b Write down the equation for the axis of symmetry of the graph The equation f x 12 has two solutions The first solution is x 10 c d e Use the symmetry of the graph to show that the second solution is x 4 om Write down the x intercepts of the graph On graph paper draw the graph of y f x for 10 x 4 and 14 y 14 Use a scale of 1 cm to represent 1 unit on the x axis and 1 cm to represent 2 units on the y axis

4 Congruence relations Recall the congruence relation of numbers congruence modulo n Let s define R4 the binary relation of congruence modulo 4 on Z Let R x y ZxZ x y mod 4 1 Calculate 0 1 2 3 4 5 You can enumerate the elements 2 How many distinct equivalence classes are there Justify your answer 3 Are the equivalence classes disjoint empty intersection What is their union

5 Domain Range Leading Coefficient Degree Number of REAL roots Number of Maxima Number of Minima 0 500 5

State all transformations 11 2 x 3 4 Factor 12 25x 49 13 25x 70X 49 by Ex 265 7 57

1 0 0081 8 2 2 5 8 5 2x 3 4 2 25 2 2 5 3 96 ti 4 24 Solve each equation Give the answer in radical form and then use a calculator to approximate solution to two decimal places if necessary 6 6x 21 33 7 6 2 24 2 6245 14 6 6 x 20 9 7x 10 18

the values of the following polynomials for the given values of x 1 P x x 3x x 8 find P 2 2 f x x 2x x 11 find f 5 3 g x x5 2x4 3x 2x 1 find g 3 4 f x x 4x x 6 find f 2 2 5 P x 4x 4x 8x find P 1

The formula for a baseball player s slugging percentage is If a player has 15 Singles 3 Doubles 1 Triple and 2 Home Runs with 62 At Bats What is the player s slugging percentage a 0 486 b 0 565 c 0 338 1S 2D 3T 4HR AB d 0 516

Graph a line that contains the point 2 7 and has a slope of 4 7 6 5 4 3 2 N Y 6 5 4 3 2 1 2 5 64 2 3 4 5 6 7 6

Graph y 3 8 9 8 7 6 5 4 3 2 V of 8 7 6 5 A 3 4 e P 2 HE 23456789

An elite law school will charge you 38 900 00 for your first year of tuition This is due at the beginning of the year You anticipate tuition to increase by 3 99 each year going forward You will spend three years in the law program You are unsure what discount rate to use to value the opportunity You have a conservative discount rate 5 16 and an aggressive rate of return 8 25 Using the aggressive rate of return what is the value today of your law school tuition Answer Format Currency Round to 2 decimal places

A family takes out a mortgage for 251 200 00 from the local bank The loan is for 30 years of monthly payments at a 5 88 APR monthly compounding What will the family s balance be on the mortgage after 5 56 years Answer Format Currency Round to 2 decimal places

In the surveys of the credibility of makeup products according to users statistics will be used in this IA to evaluate and measure the patterns that are present in makeup reviews Descriptive statistics will be used to summarize and present the significant characteristic features of the different features Hypothesis testing will be used within the investigation to understand whether certain observed trends and patterns are significant so that it is possible to determine if some makeup products were positively or negatively reviewed throughout all of them It is possible to use regression analysis to identify correlations between certain product attributes and user satisfaction thus developing a more precise perspective on credibility factors I will be providing a framework that will allow the reader to understand the trustworthiness of beauty products when new reviews become available This adaptive approach enables the fine tuning nature of evaluations with changing datasets Finally statistical methods will not only help quantify the validity of makeup products but also increase the reliability and rigor embedded in our conclusions hence making this potential discipline more evidence driven