1. Section 3.1
Power Functions

2. Objectives:
1. To define and evaluate power
functions.
2. To define even and odd functions.
3. To graph power functions and
identify the domain and range.

3. Definition
Power function A function of the form f(x) = Cxn where C, n  {real numbers}.

4. Notice that the definition includes functions in which n is rational or irrational. We will only be looking at functions with positive integral exponents. This means they will be polynomial functions and require you to use your knowledge of polynomials.

5. Power functions differ from polynomial functions in that they only have one term, and exponents can be any real number. Polynomial functions can have only non-negative integer exponents.

6. EXAMPLE 1 For f(x) = -2x4 evaluate f(-1), f(0), f(1/2), f(3)
EXAMPLE 1 For f(x) = -2x4 evaluate f(-1), f(0), f(1/2), f(3). Find the degree, the domain, the range, and graph the function.
f(-1) = -2(-1)4 = -2(1) = -2
f(0) = -2(0)4 = -2(0) = 0
f(1/2) = -2(1/2)4 = -2(1/16) = -1/8
f(3) = -2(3)4 = -2(81) = -162

7. EXAMPLE 1 For f(x) = -2x4 evaluate f(-1), f(0), f(1/2), f(3)
EXAMPLE 1 For f(x) = -2x4 evaluate f(-1), f(0), f(1/2), f(3). Find the degree, the domain, the range, and graph the function.
The degree is 4
D = {real numbers}
R = {y|y  0}

8. EXAMPLE 1 For f(x) = -2x4 evaluate f(-1), f(0), f(1/2), f(3)
EXAMPLE 1 For f(x) = -2x4 evaluate f(-1), f(0), f(1/2), f(3). Find the degree, the domain, the range, and graph the function. 1 2 3 -1 -2 -3 -4 -5

9. EXAMPLE 2 Graph g(x) = x3. Give the domain and range.4 -2 -4 6
D = {real numbers}
R = {real numbers}

10. All equation of the form f(x) = Cxn are functions (passing the vertical line test) with domain D = {real numbers}. A parabola also always has line symmetry.

11. Practice: Graph f(x) = -x4. Give the domain and range.
Roots: x = 0, multiplicity 4
f(1) = - (1)4 = - 1
f(-1) = - (-1)4 = - 1
y-axis symmetry (line symmetry)

12. Domain: all real numbers Range: (-, 0]x y
Domain: all real numbers
Range: (-, 0]

13. Practice: Graph f(x) = 2x3. Give the domain and range.
Roots: x = 0 multiplicity 3
f(1) = 2(1)3= 2
f(-1) = 2(-1)3 = -2
origin symmetry (point symmetry)

14. Domain: all real numbers Range: all real numbersx y
Domain: all real numbers
Range: all real numbers

15. Definition
Even function A function is even if and only if f(x) = f(-x),  x  Df.

16. Definition
Odd function A function is odd if and only if f(-x) = -f(x),  x  Df.

17. Power functions of even degree are even functions and power functions of odd degree are odd functions.

18. One special function is the identity function, y = x
One special function is the identity function, y = x. This is a power function of degree 1. The identity function is an odd function.

19. EXAMPLE 3 Determine whether the following functions are even, odd, or neither.
f(x) = x3
f(-x) = (-x)3 = -x3
-f(x) = -(x3) = -x3
Since f(-x) = -f(x) the function is odd.

20. EXAMPLE 3 Determine whether the following functions are even, odd, or neither.
g(x) = x4 + x2
g(-x) = (-x)4 + (-x)2 = x4 + x2
-g(x) = -(x4 + x2) = -x4 – x2
Since g(x) = g(-x) the function is even.

21. EXAMPLE 3 Determine whether the following functions are even, odd, or neither.
h(x) = x2 + 2x + 5
h(-x) = (-x)2 + 2(-x) + 5 = x2 – 2x + 5
-h(x) = -(x2 + 2x + 5) = -x2 – 2x – 5
Since h(x)  h(-x)  -h(x) the function is neither even nor odd.

22. Practice: Classify the function f(x) = 2x3 – 5.
1. Even
2. Odd
3. Neither

23. Practice: Identify the domain of the function f(x) = 2x3 – 5.
1. {x|x  real numbers}
2. {y|y  -5}
3. {y|y  real numbers}
4. None of these

24. Practice: Determine whether the following functions are even, odd, or neither.
f(x) = 4x5 + 2x3 – x
f(-x) = - 4x5 - 2x3 + x
-f(x) = -(4x5 + 2x3 - x) = - 4x5 - 2x3 + x
f(-x)= - f(x)
 f(x) is odd.

25. Practice: Determine whether the following functions are even, odd, or neither.
g(x) = 3x4 - 5x2
g(-x) = 3x4 - 5x2
g(-x) = 3(-x)4 - 5(-x)2
g(x) = g(-x)
Therefore, g(x) is even.

26. Practice: Determine whether the following functions are even, odd, or neither.
h(x) = x3 - x2 + x - 1
h(-x) = (-x)3 - (-x)2 + (- x) - 1
h(-x) = - x3 - x2 - x -1
-h(x) = - x3 + x2 - x + 1
-h(x) = - (x3 - x2 + x - 1)
h(x)  h(-x)
h(-x)  -h(x)
Therefore, h(x) is neither even nor odd.

27. Homework: pp

28. ►A. Exercises
Graph each power function. Give the domain and range of each and classify as even or odd.
5. f(x) = -1/4x4

29. ►A. Exercises
Graph each power function. Give the domain and range of each and classify as even or odd.
7. y = 5/12x24

30. ►B. Exercises
Evaluate.
13. f(-3) for f(x) = -2x3

31. ►B. Exercises
Evaluate.
15. f(-17.95) for f(x) = -2.5x16

32. ►B. Exercises
For f(x) = Cxn.
16. Find f(1)

33. ►B. Exercises
For f(x) = Cxn.
17. Find f( 3) n

34. ►B. Exercises
For f(x) = Cxn.
18. Find all zeros.

35. ►B. Exercises For f(x) = Cxn.
19. What is the multiplicity of the zero?

36. ►B. Exercises
For f(x) = Cxn.
20. Give the domain of f(x).

37. ►B. Exercises
For f(x) = Cxn.
21. Give the range of f(x) if n is odd.

38. ►B. Exercises For f(x) = Cxn.
22. If n is even, on what does the range of
f(x) depend?

39. ►B. Exercises
For f(x) = Cxn.
23. Give the range of f(x) if n is even.

40. ■ Cumulative Review
28. In ABC, find C, given a = 47, b = 63, and c = 82.

41. ■ Cumulative Review
29. Is the relation in the graph a function?

42. ■ Cumulative Review
30. If sin x = 0.3, find csc x, cos (90 – x), and sec (90 – x).

43. ■ Cumulative Review
31. Solve 2x² – 5x + 7 = x(x + 1).

44. ■ Cumulative Review
32. Find the slope of the line joining (2, 7) to (4, -5).