Properties of Functions

Section 1.6

Even functions

f(-x) = f(x)

Graph is symmetric with respect to the y-axis

Odd functions

f(-x) = -f(x)

Graph is symmetric with respect to the origin

Vote: (A) Even (B) Odd (C) Neither

1. 2. 3.

Verify graphically, then algebraically

f(x) = x2 – 5

Verify graphically, then algebraically

g(x) = 2x3 – 1

Verify graphically, then algebraically

h(x) = 4x3 – x

When is a function constant?

Local maxima?

Local minima?

Increasing intervals?

Decreasing intervals?

Use a graphing utility to graph f(x) = 6x3 – 12x + 5

for -2 < x < 2

Local maxima?

Local minima?

Increasing intervals?

Decreasing intervals?

pages 66-67 (29-50)

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page 67 (51-70)

Library of Functions

Section 1.6

Domain:Nonvertical line with slope m and y-intercept bIncreases if m > 0 Decreases if m < 0 Constant if m = 0

€

f (x) = mx + b

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ℜ

Linear function

Domain:Range: bEven functionConstant over domain

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f (x) = b

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ℜ

Constant function

Domain:Range: Slope: 1y-intercept: 0Odd functionIncreasing over domain

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f (x) = x

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ℜ

Identity function

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ℜ

Domain:Range: Nonnegative x-intercept: 0y-intercept: 0Even functionDecreasing on the interval (-∞, 0)Increasing on the interval (0, ∞)

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f (x) = x 2

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ℜSquare function

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ℜ

Domain:Range: x-intercept: 0y-intercept: 0Odd functionIncreasing on the interval (-∞, ∞)

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f (x) = x 3

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ℜCube function

€

ℜ

x-intercept: 0y-intercept: 0The function is neither even nor oddDomain & range nonnegative Increasing on the interval (0, ∞)Minimum value of 0 at x = 0

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f (x) = x

Square root fx:

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ℜ

x-intercept: 0y-intercept: 0Domain & range: The function is oddIncreasing on the interval (-∞, ∞)No local minimum or maximum

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f (x) = x3

Cube root fx:

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ℜ

Domain and range: Nonzero No interceptsThe function is oddDecreasing on (-∞, 0) and (0, ∞)

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f (x) =1

x

Reciprocal fx:

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ℜ

x-intercept: 0y-intercept: 0Domain: Range: NonnegativeThe function is evenDecreasing on the interval (-∞, 0)Increasing on the interval (0, ∞)Local minimum value of 0 at x = 0

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f (x) = x

Absolute value fx:

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ℜ

€

ℜ

Greatest integer less than or equal to xDomain:Range: y-intercept: 0 x-intercepts [0, 1)Neither even nor oddConstant on [k, k + 1)

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f (x) = int(x)

Greatest integer fx:

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ℜ

€

ℑ€

f (x) = x[ ]

pages 66-68 (13-28, 71, 73-76)