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

Verify graphically, then algebraically f(x) = x 2 – 5

Verify graphically, then algebraically g(x) = 2x 3 – 1

Verify graphically, then algebraically h(x) = 4x 3 – x

When is a function constant?

Local maxima? Local minima? Increasing intervals? Decreasing intervals?

Use a graphing utility to graph f(x) = 6x 3 – 12x + 5 for -2 < x < 2 Local maxima? Local minima? Increasing intervals? Decreasing intervals?

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Library of Functions Section 1.6

Domain: Nonvertical line with slope m and y-intercept b Increases if m > 0 Decreases if m < 0 Constant if m = 0 Linear function

Domain: Range: b Even function Constant over domain Constant function

Domain: Range: Slope: 1 y-intercept: 0 Odd function Increasing over domain Identity function

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

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

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

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

Domain and range: Nonzero No intercepts The function is odd Decreasing on (-∞, 0) and (0, ∞) Reciprocal fx:

x-intercept: 0 y-intercept: 0 Domain: Range: Nonnegative The function is even Decreasing on the interval (-∞, 0) Increasing on the interval (0, ∞) Local minimum value of 0 at x = 0 Absolute value fx:

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

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