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Activity 1 - 7

Activity 1 - 7. Fill ‘er Up. y. y. y. x. x. x. 5-Minute Check on Activity 1-6. How do we test a graph to see if it is a function? What is the input to y = f(x)? What is the output to y = f(x) A function has _____ unique output for every input value.

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Activity 1 - 7

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  1. Activity 1 - 7 Fill ‘er Up

  2. y y y x x x 5-Minute Check on Activity 1-6 • How do we test a graph to see if it is a function? • What is the input to y = f(x)? • What is the output to y = f(x) • A function has _____ unique output for every input value. • Describe the graphs of the functions below: Vertical Line Test Input is x Output is y One Increasing Decreasing Constant Click the mouse button or press the Space Bar to display the answers.

  3. Objectives • Write the equation to define a function • Determine the domain and range of a function • Identify the independent and dependent variables of a function

  4. Vocabulary • Rational number – any number that can be expressed as the quotient of two integers • Irrational number – any number that cannot be expressed as the quotient of two integers • Independent variable – another name for the input variable; the variable we control • Dependent variable – another name for the output variable; the variable we don’t control

  5. Vocabulary cont • Domain – collection of all possible values of the independent variable • Practical Domain – the collection of possible values for the domain that make sense in the context of the problem • Range – the collection of all possible values of the dependent variable • Practical Range – the collection of possible values for the range that make sense in the context of the problem • Increment – the change between two consecutive values of the independent variable • Function Notation – replaces the dependent variable with f(independent variable), y = f(x)

  6. Activity You probably need to fill your car with gas more often than you would like, so you drive around looking for the best price per gallon. What two variables determine how much the fill-up will cost? Which one do you have any choice with? How empty is the tank? -- number of gallons to fill How expensive is the gas? -- cost per gallon Cost per gallon is the only one we have a little control over

  7. Activity cont Suppose the best price you find is $3.14 9/10 per gallon. Complete the following table: Is the cost of a fill-up a function of the number of gallons pumped? Explain Identify the independent and dependent variables. Write an equation relation c, cost, and g, gallons pumped for the above problem 15.75 25.19 31.49 44.09 Yes; a one-to-one relationship exists between cost and the number of gallons pumped Cost – dependent and gallons pumped – independent C = 3.149g

  8. Real Numbers Input and output values used in our course will be real numbers. A real number can be a rational number or an irrational number. Rational numbers are any real number that can be expressed as the quotient of two integers Irrational numbers are any real numbers that cannot be expressed as the quotient of two integers Rational Numbers Irrational Numbers Note: they are complementary sets (something) we will see in Stats later Both boxes are all real numbers

  9. Domain and Range We call the collection of all possible values of the independent variable, the domain. We call the collection of all possible values of the dependent variable, the range. Practical domain and ranges are always view in the context of the problem. What are the domain and range in the Activity? What are the practical domain and range? Domain: all real numbers Range: all real numbers Domain: gals  0 Range: cost  0

  10. Domain and Range - II In general, we assume the domain (all possible x-values) to be ALL REAL #’s (x  ARN) We have two exceptions to that: • We can never divide by zeroExample: f(x) = 1 / (x – 2) • We cannot have a negative under a square root or logExample: f(x) = (x – 3) For our class, we generally graph the function to determine its range (all possible y-values) For example 1: For example 2: (Use our calculator Y= f(x), then hit ZOOM 6 to graph) Domain: x  2 (would give us 1/0) Domain: x ≥ 3 (would give us + #) Range: y  0 Range: y ≥ 0

  11. y Gross Pay Function I: hours D: pay If you work for an hourly wage, your gross pay is a function of the number of hours you work. • What is the independent and dependent variables? • If you earn $8 per hour, complete the table: • Plot the points on the graph • Write the equation, f(n) • Evaluate f(10) and explain its meaning • What are the practical domain and range of f? x 0 24 40 56 80 96 f(n) = 8n f = pay n = hours worked f(10) = 8(10) = 80 received $80 for working 10 hours D: hours worked  0 R: pay  0

  12. Summary and Homework • Summary • Independent variables are the input (ones we control) • Dependent variables are the output • Domains are all possible independent variables of the function • Ranges are all possible dependent variables of the function • Practical domains and ranges depend on the context of the problem • Functional notation: dependent variable = f(independent variable) • Homework • pg 63-67; 1-11, 14

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