Section 3.1 Power Functions

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.

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

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.

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.

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

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}

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

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

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.

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)

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

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)

Domain: all real numbers Range: all real numbers x y Domain: all real numbers Range: all real numbers

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

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

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

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.

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.

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.

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.

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

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

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.

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.

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.

Homework: pp. 109-110

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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