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Properties of Library Functions

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Explore the properties and characteristics of various functions in the library, including linear, square, square root, cube, cube root, reciprocal, absolute value, and greatest integer functions.

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Properties of Library Functions

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  1. Section 3.4 Library of Functions; Piecewise-Defined Functions

  2. THE SQUARE ROOT FUNCTION • The x-intercept is 0. The y-intercept is also 0. • The function is neither even nor odd. • It is increasing on the interval (0, ∞). • It has a local minimum of 0 at x = 0. Properties of

  3. THE CUBE ROOT FUNCTION Properties of • The x-intercept of the graph is 0. The x-intercept of the graph is also 0. • The function is odd. • It is increasing on the interval (−∞, ∞). • It does not have a local minimum or a local maximum.

  4. THE ABSOLUTE VALUE FUNCTION Properties of • The x-intercept of the graph is 0. The y-intercept of the graph is also 0. • The function is even. • It is decreasing on the interval (−∞, 0). It is increasing on the interval (0, ∞). • It has a local minimum of 0 at x = 0.

  5. LIBRARY OF FUNCTIONS 1. Linear Function:f (x) = mx + b Domain: (−∞, ∞) Its graph is a line. Its y-intercept is b Increasing if m > 0. Decreasing if m < 0. Constant if m = 0. 2. Square Function:f (x) = x2 Domain: (−∞, ∞) Its graph is a parabola. Its y-intercept is 0 It is decreasing on (−∞, 0) and increasing on (0, ∞).

  6. 4. Square Root Function: Domain: [0, ∞) Its graph is a parabola. Its y-intercept is 0 It is increasing on the interval (0, ∞). LIBRARY (CONTINUED) 3. Cube Function:f (x) = x3 Domain: (−∞, ∞) Its graph is a parabola. Its y-intercept is 0 It is increasing on (−∞, ∞).

  7. 5. Cube Root Function: Domain: (−∞, ∞) Its y-intercept is 0 It is increasing. 6. Reciprocal Function: Domain: (−∞, 0) U (0, ∞) It has no intercepts. It is increasing on the intervals (−∞, 0) and (0, ∞). LIBRARY (CONTINUED)

  8. LIBRARY (CONTINUED) 7. Absolute Value Function:f (x) = | x | Domain: (−∞, ∞) Its y-intercept is 0. It is decreasing on the interval (−∞, 0) and increasing on the interval (0, ∞) 8. Greatest Integer Function:f (x) = int(x) Domain: (−∞, ∞) It is an example of a step function. Its y-intercept is 0. It is constant on the intervals [n, n + 1) where n is any integer.

  9. PIECEWISE-DEFINED FUNCTIONS A piecewise-defined function is a function that is defined by more than one equation. An example is

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