pre cal 1-1 lesson with notes...

Pre Cal 11 Lesson with notes 5th.notebook

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11 Functions

Daily Outcome:I can describe subsets of real numbersI can identify and evaluate functions

I can state their domains

.

When can an increase in one value cause another value to increase?

When can an increase in one value cause another value to decrease?

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Subsets of Real NumbersN =

W =

Z =

Q=

I=


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Setbuilder notation: uses the properties of the numbers in the set to define the set

{x | x > 3, x N}

The set of number x such that x is greater than 3 and x is an element of the natural numbers.

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Example 1: Describe the set of numbers using setbuilder notation.

1. {2, 3, 4, 5, 6, 7} 2. x > 17 3. all multiples of seven

4. {1, 2, 3, 4, 5, . . . } 5. x ≤ 3 6. all multiples of π

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Interval notation: uses inequalities to describe subsets of real numbers.

[ or ] used to indicate that an endpoint is included in the interval( or ) used to indicated that an endpoint is not included in the

interval

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Example 2: Write each set of numbers using interval notation.

7. 2 ≤ x ≤ 12 8. x > 4 9. x < 3 or x ≥ 54

10. 4 ≤ y < 1 11. a ≥ 3 12. x > 9 or x < 2

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Relation: a rule that relates two quantities, it pairs the elements in a set A with elements in a set B.

Domain: set A of all input valuesRange: set B of all the outputs

4 common ways to represent relations

1. verbally: by a sentence that describes how the input variable is related to the output variable (The output value is 2 more than the input value)

2. numerically: by a table or list of ordered pairs that matches input values with output values

{(0, 2), (1, 3), (2, 4), (3, 5)}

3. graphically: by points on a graph in a coordinate plane in which the input values are represented by the horizontal axis and the output values are represented by the vertical axis

4. algebraically: by an equation in two variablesy = x + 2

Independent variable: the input values (usually x)Dependent variable: the output values (usually y)


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Function: a special type of relation where each x is paired with exactly one y.

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Example 3: Determine whether each relation is a function.

13. 14.2

1

012

3

4

5

Input Value 0 1 2 1 0

Output Value 4 2 0 2 4

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Determine whether each relation represents y as a function of x.

15. The input value x is the height of a student in inches, and the output value y is the number of books that the student owns.

16. The input value x is the area code, and the output value y is a phone number in that area code.


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Which sets of ordered pairs represent function(s) from A to B?

17. A = {a, b, c} and B = {0, 1, 2, 3}

(a) {(a, 1), (c, 2), (c, 3), (b, 3)}(b) {(a, 1), (b, 2), (c, 3)}(c) {(1, a), (0, a), (2, c), (3, b)}(d) {(c, 0), (b, 0), (a, 3)}

18. x y

1 1

1 1

4 2

4 2

9 3

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Determine whether each relation represents y as a function of x.

19. 20. 21.

22. 3y + 6x = 18

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Determine if the equation represents y as a function of x.25. x = y2 26. x = y + 5 27. y = |4 x|

28. 29. y = 2

To test for functions written algebraically:Solve for the dependent variable and verify that for each independent variable (x) there can be only one dependent variable (y). 23. x2 + y = 1 24. x = 3y2


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Function notation: f(x) means the value of the function f at x, it is read f of x. Because f(x) corresponds to the yvalue of f for a given xvalue, you can write y = f(y).

Equation Functiony = 4x + 3 f(x) = 4x + 3


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Example 4: Find each function value.

30. f(x) = x2 2x 8 (a) f(3) (b) f(3d) (c) f(2a 1)

31. g(y) = 7 3y (a) g(0) (b) g(7/3) (c) g(s + 2)

32. f(x) = (a) f(8) (b) f(1) (c) f(x 8)

33. (a) f(12) (b) f(6x) (c) f(3a + 8)

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Domain: all values of the independent variable for which the function is defined (x's).

Two cases to look for:1. when an x value would result in division by 0 2. when an x value would make a negative under a square root.

Implied domain: all real numbers for which the expression is defined.

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Example 5: Determine the domain of the function.

34. g(x) = 1 2x2 35. s(y) =

36. f﴾x﴿ = 37.

38. 39.

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Piecewise function: function that is defined using two or more equations for different intervals of the domain.


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Example 6:

40. Realtors in a metropolitan area studied the average home price per square foot as a function of total square footage. their evaluation yielded the following piecewisedefined function. Find the average price per square foot for a home with the given square footage.

a. 1400 square feet

b. 3200 square feet

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Determine the value of the function for each value of x.

41. f(x) = x ≤ 1 (a) f(2) (b) f(1) (c) f(2) x > 1

42. if 0 ≤ t ≤ 15 (a) v(5) (b) v(15) if 15 < t < 240 if 240 ≤ t ≤ 250 (c) v(20) (d) v(245)


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Find g(4), g(h), g(x + h), and (the difference quotient) and simplify your answer.

43. g(x) = 3x 1

pre cal 1-1 lesson with notes...

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