properties of functions section 1.6. even functions f(-x) = f(x) graph is symmetric with respect to...
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Properties of Functions
Section 1.6
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Even functions
f(-x) = f(x)
Graph is symmetric with respect to the y-axis
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Odd functions
f(-x) = -f(x)
Graph is symmetric with respect to the origin
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Vote: (A) Even (B) Odd (C) Neither
1. 2. 3.
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Verify graphically, then algebraically
f(x) = x2 – 5
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Verify graphically, then algebraically
g(x) = 2x3 – 1
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Verify graphically, then algebraically
h(x) = 4x3 – x
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When is a function constant?
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Local maxima?
Local minima?
Increasing intervals?
Decreasing intervals?
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Use a graphing utility to graph f(x) = 6x3 – 12x + 5
for -2 < x < 2
Local maxima?
Local minima?
Increasing intervals?
Decreasing intervals?
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pages 66-67 (29-50)
=======================
page 67 (51-70)
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Library of Functions
Section 1.6
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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
€
ℜ
Linear function
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Domain:Range: bEven functionConstant over domain
€
f (x) = b
€
ℜ
Constant function
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Domain:Range: Slope: 1y-intercept: 0Odd functionIncreasing over domain
€
f (x) = x
€
ℜ
Identity function
€
ℜ
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Domain:Range: Nonnegative x-intercept: 0y-intercept: 0Even functionDecreasing on the interval (-∞, 0)Increasing on the interval (0, ∞)
€
f (x) = x 2
€
ℜSquare function
€
ℜ
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Domain:Range: x-intercept: 0y-intercept: 0Odd functionIncreasing on the interval (-∞, ∞)
€
f (x) = x 3
€
ℜCube function
€
ℜ
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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
€
f (x) = x
Square root fx:
€
ℜ
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x-intercept: 0y-intercept: 0Domain & range: The function is oddIncreasing on the interval (-∞, ∞)No local minimum or maximum
€
f (x) = x3
Cube root fx:
€
ℜ
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Domain and range: Nonzero No interceptsThe function is oddDecreasing on (-∞, 0) and (0, ∞)
€
f (x) =1
x
Reciprocal fx:
€
ℜ
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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
€
f (x) = x
Absolute value fx:
€
ℜ
€
ℜ
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Greatest integer less than or equal to xDomain:Range: y-intercept: 0 x-intercepts [0, 1)Neither even nor oddConstant on [k, k + 1)
€
f (x) = int(x)
Greatest integer fx:
€
ℜ
€
ℑ€
f (x) = x[ ]
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pages 66-68 (13-28, 71, 73-76)