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Chapter 2

Chapter 2. Linear Functions and Relations. In Chapter 2, You Will…. Move from simplifying variable expressions and solving one-step equations and inequalities to working with two variable equations and inequalities.

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Chapter 2

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  1. Chapter 2 Linear Functions and Relations

  2. In Chapter 2, You Will… • Move from simplifying variable expressions and solving one-step equations and inequalities to working with two variable equations and inequalities. • Learn how to represent function relationships by writing and graphing linear equations and inequalities. • By graphing data and trend lines, you will understand how the slope of a line can be interpreted in real-world situations.

  3. 2-1 Relations and Functions What You’ll Learn … • To graph relations. • To identify functions.

  4. [(-2,4), (3,-2), (-1,0), (1,5)] [(0,4),(-2,3),(-1,3),(-2,2),(1,-3)] A relation is a set of pairs of input and output numbers. Example 1 Graphing a Relation

  5. The domain of a relation is the set of all inputs, or x-coordinates of the ordered pairs. The range of a relation is the set of all outputs, or y-coordinates of the ordered pairs. (2,4),(3,4.5),(4,7.5),(5,7),(6,5),(6,7.5) D= _____________ R= _____________ Finding Domain and Range D= _____________ R= _____________

  6. Using a Mapping (-2,-1) (-1,-1) (-2,1) (6,3) -2 -1 Another way to show a relation is to use a mapping diagram, which links elements of the domain with corresponding elements of the range. -1 0 1 6 3

  7. (0,2) (1,3) (2,4) (2,8) (-1,5) (0,8) (-1,3) (-2,3) Example 3 Making a Mapping Diagram

  8. Are the x's different? A function is a relation that assigns exactly one value in the range to each value in the domain

  9. Example 4 Identifying Functions -2 0 5 -1 3 4 -1 0 2 3 -1 3 5

  10. One way you can tell whether a relation is a function is to analyze the graph of the relation using the vertical line test. If any vertical line passes through more than one point of the graph, the relation is NOT a function.

  11. Which are Functions? • • • • • •

  12. Function Notation • Another way to write a function y = 3x + 4 is f(x)= 3x + 4. • You read f(x) as “f of x” or “f is a function of x”. • The notations g(x) and h(x) also indicate functions of x.

  13. A functionrule is an equation that describes a function. You can think of a function rule as an input-output machine Evaluating Functions Function Rule Input Output

  14. Evaluating a Function Function Rule y = 3x + 4 3x + 4 x y

  15. f(n)= -3n – 10 Find f(6). g(x) = -2x² + 7 Find g(6). Evaluating a Function Rule

  16. Example 6 Real World Connection The area of a square tile is a function of the length of a side of the square. Write a function rule for the area of a square. Evaluate the function for a square tile with side length 3.5 in.

  17. 2.01 Use the composition of functions to model and solve problems; justify results. What you’ll learn … To add, subtract, multiply and divide functions To find the composite of two numbers 7-6 Function Operations

  18. Function Operations • Addition (f+g)(x) = f(x)+g(x) • Multiplication (fg)(x) = f(x) g(x) • Subtraction (f-g)(x) = f(x) – g(x) • Division (x)= , g(x)≠0 f g f(x) g(x)

  19. Let f(x) = 3x +8 and g(x) = 2x-12. Find f+g and f - g and their domain. Let f(x) = 5x2 - 4x and g(x) = 5x+1. Find f+g and f - g and their domain. Example 1 Adding and Subtracting Functions

  20. Let f(x) = x2 - 1 and g(x) = x+1. Find fg and and their domain. Let f(x) = 6x2 +7x - 5 and g(x) = 2x-1. Find fg and and their domain. Example 2 Multiplying and Dividing Functions f g f g

  21. Composition of Functions The composition of function g with function f is written as g°f and is defined as (g°f)(x)= g(f(x)), where the domain of g°f consists of the values a in the domain of f such that f(a) is in the domain of g. (g°f)(x) = g(f(x)) • Evaluate the inner function • f(x) first. 2. Then use your answer as the input of the outer function g(x).

  22. Let f(x) = x-2 and g(x) = x2. Find (g°f)(-5). Let f(x) = x-2 and g(x) = x2. Find (f°g)(x) and evaluate (f°g)(-5). Example 3 Composition of Functions

  23. Suppose you are shopping in the store in the photo. You have a coupon worth $5 off any item. Use functions to model discounting an item by 20% and to model applying the coupon. Use a composition of your two functions to model how much you would pay for an item if the clerk applies the discount first and then the coupon. Use a composition of your two functions to model how much you would pay for an item if the clerk applies the coupon first and then the discount. How much more is any item if the clerk applies the coupon first? Example 4a Real World Connection

  24. A store is offering a 10% discount on all items. In addition, employees get a 25% discount. Write a composite function to model taking the 10% discount first. Write a composite function to model taking the 25% discount first. Suppose you are an employee. Which discount would you prefer to take first? Example 4b Real World Connection

  25. 2.01 Use the composition of functions to model and solve problems; justify results. What you’ll learn … To find the inverse of a relation or function. 7-7 Inverse Relations and Functions

  26. The Inverse of a Function If a relation maps element a of its domain to element b of its range., the inverse relation “undoes” the relation and maps b back to a. Relation r Inverse of r 1 2 1.2 1.4 1.6 1.9 1.2 1.4 1.6 1.9 1 2

  27. Find the inverse of relation s. Graph s and its inverse. Example 1 Finding the Inverse of a Relation

  28. Find the inverse of y = x2 + 3. Does y = x2 + 3 define a function? Is its inverse a function? Explain. Find the inverse of y = 3x - 10. Is its inverse a function? Explain. Example 2 Interchanging x and y

  29. Graph y= x2 + 3 and its inverse, y = +√x -3 . Graph y= 3x-10 and its inverse. Example 3 Graphing a Relation and Its Inverse

  30. The inverse of a function is denoted by f-1. Read f-1 as “the inverse of f” or as “f inverse”. The notation f(x) is used for functions, but f-1(x) may be a relation that is not a function.

  31. Example 4a Finding an Inverse Function Consider the function f(x) = √x+1. • Find the domain and range of f. • Find f-1. • Find the domain and range of f-1. • Is f-1 a function? Explain.

  32. Example 4b Finding an Inverse Function Consider the function f(x) = 10 – 3x. • Find the domain and range of f. • Find f-1. • Find the domain and range of f-1. • Find f-1(f(3)). • Find f-1(f(2)).

  33. Composite of Inverse Functions • If f and f-1 are inverse functions then, (f-1°f)(x) and (f°f-1)(x) = x.

  34. Example 6 Composite of Inverse Functions • For f(x) = 5x + 11, find (f-1°f)(777). • For f(x) = 5x + 11, find (f°f-1)(-5802).

  35. What you’ll learn … To graph linear equations. To write equations of lines. 2-2 Linear Equations

  36. A function whose graph is a line is a linear function. You can represent a linear function with a linear equation, such as y=3x+2. A solution is any ordered pair (x,y) that makes the equation true. Because the y depends on the value of x, y is called the dependent variable and the x is called the independent variable. Graphing Linear Equations

  37. Graph the equations using a table. y=-3x y=½x+3 Example 1 Graphing a Linear Equation

  38. The y intercept of a line is the point in which the line crosses the y-axis. • The x intercept of a line is the point in which the line crosses the x-axis. • The standard form of a linear equation is Ax +By = C, where A,B and C are real numbers and A and B are not both zero.

  39. The equation 3x +2y =120 models the number of passengers who can sit in a train car, where x is the number of adults and y is the number of children. Graph the equation. Describe the domain and range. Explain what the x and y intercepts represent. Example 2 Real World Connection

  40. Slope • The slope of a non-vertical line is the ratio of the vertical change to a corresponding horizontal change. • You can calculate the slope by subtracting the corresponding coordinates of two points on the line.

  41. Slope Formula Vertical change (rise) Horizontal change (run) y2 – y1 x2 – x1 =

  42. Find the slope of the line through the points (3,2) and (-9,6). Find the slope of the line through the points (5,2) and (-6,2). Example 3 Finding Slope

  43. Point-Slope Form • When you know the slope and a point on a line, you can use the point-slope form to write an equation of the line. y – y1 = m (x - x1)

  44. Write in standard form an equation of the line with slope -½ through the point (8,-1). Write in standard form an equation of the line with slope 2 through the point (4, -2). Example 4 Writing an Equation Given the Slope and a Point

  45. Write in point slope form an equation of the line through (1,5) and (4,-1). Write in point slope form an equation of the line through (-2,-1) and (-10,17). Example 5 Writing an Equation Given Two Points

  46. Slope Intercept Form • Another form of the equation of a line is slope intercept form, which you can use to find the slope by examining the equation. y= mx +b Slope y intercept

  47. Find the slope of 4x + 3y = 7. Find the slope of ½x + ¾y = 1 Example 6 Finding Slope Using Slope-Intercept Form

  48. Summary: Equations of Lines

  49. Special Slopes Vertical Line Horizontal Line Zero Slope Undefined Slope

  50. Special Slopes Perpendicular Lines Parallel Lines Have same slopes Have reciprocal slopes

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