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Warm Up

Warm Up. 1. If a² = 12, then a 4 = A. 144 B. 72 C. 36 D. 24 E. 16 2. If n is even, which of the following cannot be odd? I n + 3 II 3n III n² - 1 A. I only B. II only C. III only D. I and II only E. I, II and III. Chapter 1: Functions and their Graphs. 1.1 Functions. Objectives.

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Warm Up

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  1. Warm Up • 1. If a² = 12, then a4 = A. 144 B. 72 C. 36 D. 24 E. 16 • 2. If n is even, which of the following cannot be odd? I n + 3II 3nIII n² - 1 A. I only B. II only C. III only D. I and II only E. I, II and III

  2. Chapter 1: Functions and their Graphs 1.1 Functions

  3. Objectives Today: • SWBAT identify a function. • SWBAT use function notation to evaluate functions. Tomorrow: • SWBAT find the domain of functions. • SWBAT use functions to model and solve real-life problems.

  4. Vocabulary • Relation: pairs of quantities that are related to each other by some rule of correspondence. • Example: The area A of a circle is related to its radius r by the formula A = π r2 • Function: A relation that matches each input with exactly one output. • Can be represented by a sentence, a table, a graph, or an equation. • Input, domain, x-value, independent variable • Output, range, y-value, dependent variable

  5. Identifying Functions • Read the sentence, examine the graph, table or equation. • Look for repeated inputs. • If all inputs are different, the relation is a function. • If an input is repeated and the output is different, the relation is NOT a function.

  6. Example 1 1. The input value x is the number of representatives from a state, and the output value y is the number of senators. • Solution: Y is a function of x. Regardless of the value of x, the value of y is always 2. These are called constant functions.

  7. Example 2 2. Solution: Y is NOT a function of x. The input value 2 is mapped to two different y – values (11 and 10).

  8. Example 3 3. Solution: Y is a function of x. No input value is mapped to two output values. You can determine this through the vertical line test.

  9. Example 4 4. x2 + y = 1 • Hint: To determine whether y is a function of x, try to solve for y in terms of x. • Solution: To each value of x there corresponds exactly one value of y. So, y is a function of x.

  10. Example 5 5. –x + y2 = 1 • The ± indicates that for a given value of x there are two corresponding values of y. So, y is NOT a function of x.

  11. Practice Determine if each relation represents y as a function of x. • The input value x is the year, and the output value y is the circulation of newspapers. • x = y2 • y = x2 + 5 • x + y2 = 4

  12. Practice

  13. Practice 1. 2.

  14. Function Notation • A convenient way to name functions so that they can be referenced easily. • f(x) is read as the “value of f at x” or “f of x”. • f(x) = y

  15. Evaluating Functions with Function Notation • Function Values are denoted by f(-1), f(0), f(2), etc. • To find these values, substitute the specified input values into the given equation. For x = -1, f(-1) = For x = 0, f(0) = For x = 2, f(2) =

  16. Examples • g(x) = x2 + 4x + 1, find g(t) and g(2) • s(r) = r3 + r – 5, find s(a) and s(-2)

  17. Practice Evaluate the function at each specified value of the independent variable. • g(y) = 7 – 3y, g(0) and g(⅔) • f(x) = 15 – 3x, f(5) and f(-2)  • h(t) = t2 – 2t, h(2) and h(1.5) • s(t) = t3 – 8, s(x + 2) and s(-2) • , q(2) and q(-x) 

  18. Piecewise Functions • A piecewise function is a type of function whose domain is defined in pieces. • Evaluate the function when x = -1, 0, and 1.

  19. Independent Practice/Homework • Work with a partner/ share a book • p. 82 # 2 – 24 even and p. 83 # 28 – 38 even

  20. Warm Up • Does the equation represent y as a function of x? 1. 2. y = |4 – x| • Evaluate the function at each specified value of the independent variable and simplify. 3. a) q(0) b) q(3) c) q(y + 3)

  21. Interval Notation • Interval Notation: • A connected subset of numbers. • An alternative way of expressing an inequality. • When using interval notation, the symbol: • ( means "not included" or "open". • [ means "included" or "closed". • Example: • 2 < x < 6 (as an inequality) • [2, 6) (in interval notation)

  22. Interval Notation • Non-ending Interval: Infinity is included in the interval • The sign ∞ is used to represent infinity • Examples: • Infinity is always in an open interval

  23. The Domain of a Function • Domain: The set of all values of the independent variable for which the function is defined (or for which the function makes sense). • Can be explicitly defined or implied by the expression used to define the function. • Implied Domain: the set of all real numbers for which the expression is defined.

  24. The Domain of a Function • Identify the independent variable. • If given points, list all the x-values. • If given an equation: • The domain is all real numbers unless you have a root or a fraction. • If you have an even root, any values that make the root negative will be excluded from the domain. • If you have division, any values that make the denominator zero will be excluded from the domain.

  25. Examples 1. f: {(-3, 0), (-1, 4), (0, 2), (2, 2), (4, -1)}

  26. Example 2.

  27. Example 3. Volume of a sphere:

  28. Example 4.

  29. Practice

  30. Applications of Functions( …why functions are important!!) • Functions are used to model a variety of information – everything from profits in business to natural phenomena such as force. • Example 1: • You work in the marketing department of a soft-drink company and are experimenting with a new soft drink can that is slightly narrower and taller than a standard can. For your experimental can, the ratio of the height to the radius is 4. • Express the volume of the can as a function of the radius r. • Express the volume of the can as a function of the height h. • Hint: the volume of a right circular cylinder is V = πr2h

  31. Example 2 • A baseball is hit at a point 3 feet above ground at a velocity of 100 feet per second and an angle of 45 degrees. The path of the baseball is given by the function f(x) = -0.0032x2 + x + 3 where x and y are measured in feet. Will the baseball clear a 10-foot fence located 300 feet from home plate?

  32. Difference Quotient • One of the basic definitions in calculus uses the following ratio called the difference quotient:

  33. Evaluating the Difference Quotient • Example 1: • For f(x) = -2x + 4, find

  34. Evaluating the Difference Quotient • Example 2: • For f(x) = x2 – 4x + 7, find

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