1. Sec. 1.4

2.  Determine whether relations between two variables are functions; Use function notation.  Find the domains of functions.  Use functions to model & solve real-life problems.  Evaluate difference quotients.

4.  A rule of correspondence for two quantities  Math examples: equations & formulas

5.  A relation that assigns to each input exactly one output  Input  x  domain  independent variable  Output  y  range  dependent variable  Ways to represent a function:  Verbally  Numerically  Graphically  Algebraically

6. Determine whether the relation represents y as a function of x. a. The input value x is the number of representatives from a state, and the output value y is the number of senators - Verbal representation - Regardless of x, the y value is always 2. - So it is a constant function.

7. Determine whether the relation represents y as a function of x. b. xy 211 210 38 45 51 Numerical representation {(2, 11), (2, 10) (3, 8), (4, 5), (5, 1)} It is NOT a function; x repeats.

8. Determine whether the relation represents y as a function of x. c. - Graphical representation - It is a function.

9. Determine whether the relation represents y as a function of x.

10. Which of the equations represent(s) y as a function of x? a. x 2 + y = 1 b. -x + y 2 =1

11. Which of the equations represent(s) y as a function of x? a. x 2 + y = 1 b. -x + y 2 =1 Solve for y! x 2 + y = 1 y = - x 2 + 1 -x + y 2 =1 y 2 = x + 1 y = ± √x + 1

12. Which of the equations represent(s) y as a function of x? a. x 2 + y 2 = 8 b. y – 4x 2 = 36 No Yes

13.  Naming a function  Especially an equation representing a function f(x)f(x)  The value of f at x  f of x yy

14. Let g(x)= -x 2 + 4x + 1. Find each function value. a. g(2) b. g(t) a. g(x+2)

15. Let g(x)= -x 2 + 4x + 1. Find each function value. a. g(2) b. g(t) a. g(x+2) g(x)= -x 2 + 4x + 1 g(2)= -(2) 2 + 4(2) + 1 g(2)= -4 + 8 + 1 g(2)= 5

16. Let g(x)= -x 2 + 4x + 1. Find each function value. a. g(2) b. g(t) a. g(x+2) g(x)= -x 2 + 4x + 1 g(t)= -(t) 2 + 4(t) + 1 g(t)= -t 2 + 4t + 1

17. Let g(x)= -x 2 + 4x + 1. Find each function value. a. g(2) b. g(t) a. g(x+2) g(x)= -x 2 + 4x + 1 g(x+2)= -(x+2) 2 + 4(x+2) + 1 g(x+2)= -(x 2 +4x+4) + 4x+8 + 1 g(x+2)= -x 2 -4x-4 + 4x+8 + 1 g(x+2)= -x 2 + 5

18. Let f(x)= 10 – 3x 2. Find each function value. a. f(2) b. f(-4) a. f(x – 1) -2 -38 -3x 2 + 6x + 7

19.  A function defined by two or more equations over a specific domain

20. Evaluate the function when x= -1, 0, and 1.

21. f(-1)= (-1) 2 + 1 f(-1)= 1 + 1 f(-1)= 2 f(0)= (0) – 1 f(0)= -1 f(1)= (1) – 1 f(1)= 0

22. Evaluate the function when x= -2, 2, and 3.

23. f(-2)= (-2) 2 + 1 f(-2)= 4 + 1 f(-2)= 5 f(2)= (2) – 1 f(2)= 1 f(3)= (3) – 1 f(3)= 2

24. Find all real values of x such that f(x) = 0. a. f(x)= -2x + 10 a. f(x)= x 2 – 5x + 6

25. Find all real values of x such that f(x) = 0. a. f(x)= -2x + 10 a. f(x)= x 2 – 5x + 6 -2x + 10 = 0 -2x = -10 x = 5 x 2 – 5x + 6 = 0 (x – 2)(x – 3) = 0 x – 2 = 0x – 3 = 0 x = 2 x = 3 You can also use the quadratic formula to solve this equation.

26. Find all real values of x such that f(x) = 0. f(x)= x 2 – 16

27. Find all real values of x such that f(x) = 0. f(x)= x 2 – 16 x 2 – 16 = 0 x 2 = 16 x = ±4

28. Find the values of x for which f(x) = g(x). a. f(x)= x 2 + 1 and g(x)= 3x – x 2 a. f(x)= x 2 – 1 and g(x)= -x 2 + x + 2

29. Find the values of x for which f(x) = g(x). a. f(x)= x 2 + 1 and g(x)= 3x – x 2 x 2 + 1 = 3x – x 2 2x 2 – 3x + 1 = 0 (x – 2)(x – 1) = 0 (x – )(x – ½) = 0 (x – 1)(2x – 1) = 0 x – 1 = 0 2x – 1 = 0 x = 1 2x = 1 x = ½

30. Find the values of x for which f(x) = g(x). b. f(x)= x 2 – 1 and g(x)= -x 2 + x + 2 x 2 – 1 = -x 2 + x + 2 2x 2 – x – 3 = 0 (x + 2)(x – 3) = 0 (x + )(x – ) = 0 (x + 1)(2x – 3) = 0 x + 1 = 0 2x – 3 = 0 x = -1 2x = 3 x =

31. Find the values of x for which f(x) = g(x), where f(x)= x 2 + 6x – 24 and g(x)= 4x – x 2.

32. f(x) = g(x) x 2 + 6x – 24 = 4x – x 2 2x 2 + 2x – 24 = 0 x 2 + x – 12 = 0 (x – 3) (x + 4) = 0 x = 3 x = -4

33.  The x values  Implied Domain – the set of all real numbers for which the expression is defined

34. Find the domain of each function. a. f: {(-3, 0), (-1, 4), (0, 2), (2, 2), (4, -1)} The domain of f consists of all first coordinates in the set of ordered pairs. Domain = {-3, -1, 0, 2, 4}

35. Find the domain of each function. b. The denominator cannot equal zero. So you must exclude those x-values. Domain = all real numbers x except x=-5 x + 5 = 0 x = -5

36. Find the domain of each function. c. Volume of a sphere: This is a real-world example (volume). So the dimensions (r) must be positive. Domain = all real numbers such that r > 0

37. Find the domain of each function. d. The radicand must be positive… Cannot take the square root of a negative value. Domain = all real numbers x ≤ 4 – 3x ≥ 0 -3x ≥ -4 x ≤

39. One of the basic definitions in calculus employs the ratio h ≠ 0

40. For f(x)= x 2 – 4x + 7, find