Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Precalculus Pretest- Unit 1 Name:______________________________ Date:_______________________________ MULTIPLE CHOICE (INDICATE YOUR ANSWERS IN THE SPACE PROVIDED): (1) Which graph illustrates a quadratic relation whose domain is all real numbers? (a) (2) (b) (c) (1) (d) Which graph of a relation is also a function? (2) (a) (3) (b) (c) (d) Which function is not one to one? (3) (a) (4) (c) (d) If the following graph is y = f(x), what is the value of f(1)? (a) -1 (5) (b) (b) -2 For which value(s) of x is the function f ( x ) (4) (c) 1 (d) 2 x2 9 undefined? x 7 (5) (a) 9 (b) 7 (c) 3 (d) 3 and -3 1 (6) What is the domain of f ( x ) x 6 ? (6) (a) x 6 (7) (c) x > 6 (d) x = 6 Given the relation {(-2, 3), (a, 4), (1, 9), (0, 7)}, which replacement for a makes this relation a function? (a) 1 (8) (b) x 6 (b) -2 (7) (c) 0 (d) 4 Which of the following is not a linear function? (8) (a) x = 8 (9) (b) 2x + 3y = 6 (c) y = -2x – 1 (d) y = 8 If f(x) = 2x and g(x) = x – 4, what is the value of f(g(3))? (9) (a) 2 (b) -6 (c) -2 (d) 6 (10) If f ( x ) 3x 2 and g( x ) 2x , what is the value of f g8 ? (10) (a) 16 (b) 8 6 (c) 144 (d) 48 (11) A line passes through the points (-1, 4) and (2, 2). Find the slope of any line perpendicular to this line: (a) 3 2 (b) 2 3 (c) (11) 3 2 (d) 2 3 (12) Given the set A = {(1, 2), (2, 3), (3, 4), (4, 5)}, if the inverse of the set is A-1, which statement is true? (12) (a) A and A-1 are functions. (b) A and A-1 are not functions. (c) A is a function and A-1 is not a function. (d) A is not a function and A-1 is a function. (13) What is the inverse of the equation y = 2x – 5? (13) (a) y = 2x + 5 (b) y 1 x 5 2 (c) y = 5 – 2x (d) y 1 x 5 2 2 Unit 1: Graphs & Functions Lesson Homework Review Complete pages 4-8 in the packet (2 days) Difference HW #4: p. 145: #77 – 80* Quotient Symmetry & HW #5: p. 112: #33, 34, 36, 38 – 45, 48* Odd/Even/Neither Functions Test Review Problem Set beginning on p 3 Precalculus Review Material: Functions and Graphs Name:______________________________ Date:_______________________________ Objective: To work in groups to rediscover the properties of functions and their graphs from Math 3. LINEAR FUNCTIONS: Forms of Linear Equations: slope-intercept form: standard form: point-slope form PBLM SET 1. Write the linear equation in slope intercept, standard, and point-slope form given the line passes through (5, 2) and (7, 9) 2. Write the equation of the horizontal line that passes through (-9, 2) Parallel & Perpendicular Lines Parallel lines have __________ slopes. Perpendicular lines have slopes that are ____________ ______________. 3. Write the linear equation in standard form given that the line passes through (-2, 10) and is 4 parallel to the graph of y 3x 5 4. Write the equation of the line that passes through (6, -5) and is perpendicular to the graph of 2 4 y x 3 7 4 IDENTIFYING FUNCTIONS Definitions: Relation: Function: Domain: Range: PBLM SET. 1. State the domain and range of each relation, and state whether the relation is a function or not: 2. a. {(-2, 0), (3, 2), (4, 5)} b. {(6, -2), (3, 4), (6, -6), (-3, 0)} Which relation is a function? Why? What is the shortcut rule for determining a function? (b) 3. (b) (c) (d) Find the Domain for each: a. x2 2x f ( x) x 1 b. c. f ( x) x 3 d. f(x) - x 2 2x - 27 f ( x) 1 x 4 2 5 Evaluating Functions: substitute numerical value or variable into function equation and simplify (1) find f(-1) if f(x) = x2 – 1 ____________________________________________________________________________________ (2) find h(3) if h(x) = 3x2 (3) find f(-7) if f(w) = 16 + 3w – w2 (4) find g(m) if g(x) = 2x6 – 10x4 – x2 + 5 (5) find k(w + 2) if k(x) = 3x + 4 (6) find h(a – 2) if h(x) = 2x2 – x + 3 6 Operations with Functions: given functions f and g sum: f g( x) f ( x) g( x) product: f g( x) f ( x) g( x) f f ( x) ( x ) , where g( x ) 0 g( x ) g difference: quotient: f g(x) f (x) g(x) Given functions f and g: (a) perform each of the basic operations, (b) find the domain for each (1) f ( x) 5x 4 ; g( x) x 2 1 (2) f ( x ) 5 x ; g( x ) x 1 Composition of Functions: given functions f and g notation: f g( x) f g( x) Find f g( x) and g f ( x) for each f(x) and g(x): 1. f ( x) x 2 1; g( x) 3x 2. f ( x ) x 1; g( x ) x 2 3. f ( x ) 2x 5 ; g( x ) 3 x 4. f ( x) x 2 x ; g( x) x 9 5. f ( x) 2x 3 ; g( x) x 2 2x 6. f ( x) x 1; g( x) 4x 2 7 One-to-one Functions A function is one-to-one when no two ordered pairs in the function have the same ordinate and different abscissas. The best way to check for one-to-oneness is to apply the vertical line test and the horizontal line test. If it passes both, then the function is one-to-one. (**Note: if a function is not one-to-one, it does not have an inverse**) Examples: Determine whether the following functions are one-to-one. 1) 2) 3) f ( x) x 2 1 f ( x) 2 x 1 f ( x) 3 5 x Inverse Functions Steps: 1. Write the equation in terms of x and y. 2. Switch the x with the y. 3. Solve for y. PBLM SET 1. Find the inverse of y 4 x 8 3. 2. Find the inverse of f ( x) 5 x 2 Graph each of the lines in examples one and two and their inverses on the same set of axes and identify any interesting characteristics. y x 4. 1 Given the function f ( x ) x 2 3 (a) Algebraically, find f -1(x). (b) Algebraically, verify your answer to part (a). 8 Precalculus Lesson- Evaluate functions with variables; difference quotient Objectives: Name:______________________________ Date:_______________________________ evaluate functions as expressions that involve one or more variables explore functions by evaluating and simplifying a difference quotient Evaluating & Simplifying a Difference Quotient: (1) For f(x) = x2 + 3x + 7, evaluate and simplify: (a) (2) f x h (b) f x h f x , h0 h (c) f ( x ) f (a ) , x a 0 x a For f(x) = 3x2 – 2, evaluate and simplify: (a) f x h (b) f x h f x , h0 h 9 (3) (4) For each given function, find and simplify: (a) f(x) = -x2 – 2x – 4 (b) f(x) = -2x + 5 (c) f(x) = x2 + 4x + 5 For the function f(x) = 4x – 7, find and simplify: (a) (5) f ( x h) f ( x ) , h0 h f ( x h) f ( x ) , h0 h (b) f ( x ) f (a ) , x a 0 x a (b) f ( x ) f (a ) , x a 0 x a For the function f(x) = -5x + 2, find and simplify: (a) f ( x h) f ( x ) , h0 h 10 Precalculus Lesson- Symmetry, Odd/Even/Neither Functions Name:______________________________ Date:_______________________________ Objectives: ~To learn an algebraic method for testing for symmetry with respect to axes and origin ~To learn an algebraic method for determining whether functions are even/odd/neither ~To learn shortcut approaches for the above Odd Functions symmetric with respect to the origin TEST: f(-x) = -f(x) Even Functions symmetric with respect to the y-axis TEST: f(x) = f(-x) Symmetry Tests symmetric with respect to the: y-axis x-axis origin the given equation is equivalent when: x is replaced with -x y is replaced with –y x and y are replaced with –x and -y Examples: 11 Unit 1: Graphs & Functions Definitions, Properties & Formulas Linear Equation Slope equation of a straight line the slope, m, of the line through (x1, y1) and (x2, y2) is given by the following y y1 equation, if x1 x2: m 2 x 2 x1 y Types of Slope Positive y x Negative x y-intercept where the graph crosses the y-axis x-intercept where the graph crosses the x-axis Slope-Intercept Form Standard Form Point-Slope Form Parallel Lines y y x Zero horizontal line: y=b x Undefined vertical line: x =a y = mx + b where m represents the slope and b represents the y-intercept of the linear equation Ax + By = C where A, B, and C are constants and A 0 (positive, whole number) y – y1 = m(x – x1) where m represents the slope and (x1, y1) are the coordinates of a point on the line of the linear equation Two nonvertical lines in a plane are parallel if and only if their slopes are equal 12 and they have no points in common. (Two vertical lines are always parallel.) ex) Perpendicular Lines y = 2x + 3 m=2 and y = 2x – 4 m=2 equal slopes // lines Two nonvertical lines in a plane are perpendicular if and only if their slopes are negative reciprocals. (A horizontal and a vertical line are always perpendicular.) 5 2 y x 7 y x 1 ex) and neg. recip. slopes 2 5 5 2 lines m= m= 2 5 Relation a set of ordered pairs (x, y) Domain the set of all x-values of the ordered pairs Range the set of all y-values of the ordered pairs Function a relation in which each element of the domain is paired with exactly one element in the range. Vertical Line Test (VLT) Horizontal Line Test (HLT) One-to-One Functions Inverse Relations & Functions Writing Inverse Functions If any vertical line passes through two or more points on the graph of a relation, then it does not define a function. If any horizontal line passes through two or more points on the graph of a relation, then its inverse does not define a function. a function where each range element has a unique domain element (use HLT to determine) f -1(x) is the inverse of f(x), but f -1(x) may not be a function (use HLT to determine) To find f -1(x): (1) let f(x) = y (2) switch the x and y variables (3) solve for y (4) let y = f -1(x) Odd Functions symmetric with respect to the origin TEST: f(-x) = -f(x) Even Functions symmetric with respect to the y-axis TEST: f(x) = f(-x) Symmetry Tests Operations with Functions symmetric with respect to the: y-axis x-axis origin the given equation is equivalent when: x is replaced with -x y is replaced with –y x and y are replaced with –x and -y sum: (f + g)(x) = f(x) + g(x) difference: (f – g)(x) = f(x) – g(x) 13 Composition of Functions product: (f g)(x) = f(x) g(x) quotient: f f ( x) ( x ) , where g( x ) 0 g( x ) g given functions f and g, the composite function is f g( x) f g( x) , where g(x) is substituted for x in the f(x) function Precalculus Review- Unit 1 Test Name:______________________________ Date:_______________________________ SHOW ALL WORK! (1) Which of the following equations define functions? Explain your reasoning. (a) y = x + 6 (b) y2 = x + 1 (c) y3 = x + 4 (d) y = x – 2 (e) y3 = x – 3 (f) y2 = x – 5 (2) Which of the following functions are one-to-one? Explain your reasoning. (a) f(x) = x5 (b) g(x) = 2x + 7 (c) h(x) = x2 – 4 (3) Write the linear equation in standard form Ax + By = C that passes through the points (3, 5) and (4, -3). 14 (4) Write the linear equation in slope-intercept form y = mx + b that passes through the point (1, 3) and is parallel to the line 2x + 2y = 5. (5) Write the linear equation in slope-intercept form y = mx + b that passes through the point (1, 3) and is perpendicular to the line 3x – 3y = 4. (6) Write the linear equation in standard form Ax + By = C that is a horizontal line and passes through the point (-9, 2). (7) Given f(x) = 2x2 – x + 3 find f(k – 2) 15 (8) Given f(x) = 4x – 7 and g(x) = 2x – x2, evaluate f(2) + g(-1) 16 (9) For the function f(x) = -6x + 5, find and simplify: f ( x h) f ( x ) , h0 (a) h (10) Find the domain of each of the following: 2x 3 (a) f ( x ) 2x (b) f ( x ) f (a ) , x a 0 x a (b) g( x ) x 4 (11) Algebraically determine the symmetry with respect to the y-axis, x-axis, and origin, if any exists, for each of the given equations: (a) 2x – 4y = 7 (b) 9x2 – 4y2 = 36 (12) Algebraically determine if each of the given functions are odd, even, or neither: (a) f(x) = x2 – 6x (b) f(x) = x6 + 7 17 18 (13) Perform the four basic operations with the functions f(x) = 4x and g(x) = x2 + 2 (f + g)(x) = (f – g)(x) = (f g)(x) = f ( x ) g (14) Given f(x) = x2 – 3 and g( x ) 2x 4 , find f g( x) and g f ( x) and give each domain. (15) For each of the following functions, f(x), find the inverse, f -1(x): (a) f(x) = 5x + 2 4 (b) f ( x ) x3 (16) Given the graph on the right, answer the following questions: (a) write the linear equation in slope-intercept form: (b) write in slope-intercept form the equation of the parallel line through (2, -1) and graph: (c) y x write in slope-intercept form the equation of the perpendicular line through the x-intercept and graph: 19