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Understanding Functions: Independent and Dependent Variables

Explore the concepts of independent and dependent variables in linear functions, identify inputs and outputs. Learn to determine if a given relation is a function with examples.

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Understanding Functions: Independent and Dependent Variables

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  1. Vocabulary • Dependent Variable • Independent Variable • Input • Output • Function • Linear Function

  2. Definition • Dependent Variable – A variable whose value depends on some other value. • Generally, y is used for the dependent variable. • Independent Variable – A variable that doesn’t depend on any other value. • Generally, x is used for the independent variable. • The value of the dependent variabledepends on the value of the independent variable.

  3. On a graph; the independent variable is on the horizontal or x-axis. the dependent variable is on the vertical or y-axis. Independent and Dependent Variables y dependent x independent

  4. Example: Identify the independent and dependent variables in the situation. A veterinarian must weight an animal before determining the amount of medication. The amount of medicationdepends on the weight of an animal. Dependent: amount of medication Independent: weight of animal

  5. Your Turn: Identify the independent and dependent variable in the situation. A company charges $10 per hour to rent a jackhammer. The cost to rent a jackhammerdependson the length of time it is rented. Dependent variable: cost Independent variable: time

  6. Your Turn: Identify the independent and dependent variable in the situation. Camryn buys p pounds of apples at $0.99 per pound. The cost of applesdepends on the number of pounds bought. Dependent variable: cost Independent variable: pounds

  7. Definition • Input – Values of the independent variable. • x – values • The input is the value substituted into an equation. • Output – Values of the dependent variable. • y – values. • The output is the result of that substitution in an equation.

  8. Function • In the last 2 problems you can describe the relationship by saying that the perimeter (dependent variable – y value) is a function of the number of figures (independent variable – x value). • A function is a relationship that pairs each input value with exactly one output value.

  9. Function input You can think of a function as an input-output machine. x 6 2 function y = 5x 5x 10 30 output

  10. Helpful Hint There are several different ways to describe the variables of a function. Independent Variable Dependent Variable y-values x-values Input Output Domain Range x f(x)

  11. A function is a set of ordered pairs (x, y) so that each x-value corresponds to exactly one y-value. Some functions can be described by a rule written in words, such as “double a number and then add nine to the result,” or by an equation with two variables. One variable (x) represents the input, and the other variable (y) represents the output. Function Rule y = 2x + 9 Output variable Input variable

  12. Linear Function • Another method of representing a function is with a graph. • A linear function is a function whose graph is a nonvertical line or part of a nonvertical line.

  13. Example: Representing a Linear Function A DVD buyers club charges a $20 membership fee and $15 per DVD purchased. The table below represents this situation. +15 +15 +15 +15 +15 Find the first differences for the total cost. constant linear Since the data shows a ___________ difference the pattern is __________. If a pattern is linear then its graph is a straight _________. line

  14. Relations • A relation is a mapping, or pairing, of input values with output values. • The set of input values is called the domain. • The set of output values is called the range.

  15. Domain is the set of all x values. Range is the set of all y values. Domain & Range {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)} Example 1: Domain- Range- D: {1, 2} R: {1, 2, 3}

  16. Example 2: Find the Domain and Range of the following relation: {(a,1), (b,2), (c,3), (e,2)} Domain: {a, b, c, e} Range: {1, 2, 3} Page 107

  17. Every equation has solution points(points which satisfy the equation). 3x + y = 5 (0, 5), (1, 2), (2, -1), (3, -4) Some solution points: Most equations haveinfinitely manysolution points. Page 111

  18. Ex 3. Determine whether the given ordered pairs are solutions of this equation. (-1, -4)and(7, 5);y = 3x -1 The collection of all solution points is the graph of the equation.

  19. 3.3 Functions • A relation as a function provided there is exactly one output for each input. • It is NOT a function if at least one input has more than one output Page 116

  20. In order for a relationship to be a function… Functions EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT INPUT (DOMAIN) FUNCTIONMACHINE (RANGE) OUTPUT

  21. Example 6 Which of the following relations are functions? R= {(9,10, (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)} No two ordered pairs can have the same first coordinate (and different second coordinates).

  22. Identify the Domain and Range. Then tell if the relation is a function. InputOutput -3 3 1 1 3 -2 4 Function? Yes: each input is mapped onto exactly one output Domain = {-3, 1,3,4} Range = {3,1,-2}

  23. Identify the Domain and Range. Then tell if the relation is a function. InputOutput -3 3 1 -2 4 1 4 Domain = {-3, 1,4} Range = {3,-2,1,4} Notice the set notation!!! Function? No: input 1 is mapped onto Both -2 & 1

  24. Is this a function? 1. {(2,5) , (3,8) , (4,6) , (7, 20)} 2. {(1,4) , (1,5) , (2,3) , (9, 28)} 3. {(1,0) , (4,0) , (9,0) , (21, 0)}

  25. The Vertical Line Test If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function. Page 117

  26. Use the vertical line test to visually check if the relation is a function. (4,4) (-3,3) (1,1) (1,-2) Function? No, Two points are on The same vertical line.

  27. Use the vertical line test to visually check if the relation is a function. (-3,3) (1,1) (3,1) (4,-2) Function? Yes, no two points are on the same vertical line

  28. Examples • I’m going to show you a series of graphs. • Determine whether or not these graphs are functions. • You do not need to draw the graphs in your notes.

  29. YES! Function? #1

  30. YES! Function? #2

  31. Function? #3 NO!

  32. YES! Function? #4

  33. Function? #5 NO!

  34. YES! Function? #6

  35. Function? #7 NO!

  36. Function? #8 NO!

  37. YES! #9 Function?

  38. Function? YES! #10

  39. Function? #11 NO!

  40. YES! Function? #12

  41. Function Notation “f of x” Input = x Output = f(x) = y

  42. x y x f(x) Before… Now… y = 6 – 3x f(x) = 6 – 3x -2 -1 0 1 2 12 -2 -1 0 1 2 12 (x, f(x)) (x, y) 9 9 6 6 3 3 0 0 (input, output)

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