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PRECALCULUS I

PRECALCULUS I. Functions and Graphs Function, domain, independent variable Graph, increasing/decreasing, even/odd. Dr. Claude S. Moore Danville Community College. Definition: Function.

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PRECALCULUS I

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  1. PRECALCULUS I • Functions and Graphs • Function, domain, independent variable • Graph, increasing/decreasing, even/odd Dr. Claude S. MooreDanville Community College

  2. Definition: Function A function f from set A to set B is a rule of correspondence that assigns to each element x in set A exactly one element y in set B. Set A is the domain (or set of inputs) of the function f, and set B contains range (or set of outputs).

  3. Characteristics of a Function 1. Each element in A (domain) must be matched with an element of B (range). 2. Each element in A is matched to not more than one element in B. 3. Some elements in B may not be matched with any element in A. 4. Two or more elements of A may be matched with the same element of B.

  4. Functional Notation Read f(x) = 3x - 4 as “f of x equals three times x subtract 4.” x inside parenthesis is theindependent variable. f outside parenthesis is the dependent variable. For the function f(x) = 3x - 4, f(5) = 3(5) - 4 = 15 - 4 = 11, and f(-2) = 3(-2) - 4 = - 6 - 4 = -10.

  5. Piece-Wise Defined Function A “piecewise function” defines the function in pieces (or parts). In the function below, if x is less than or equal to zero, f(x) = 2x - 1; otherwise, f(x) = x2 - 1.

  6. Definition: Function A function f from set A to set B is a rule of correspondence that assigns to each element x in set A exactly one element y in set B. Set A is the domain (or set of inputs) of the function f, and set B contains range (or set of outputs).

  7. Piece-Wise Defined Function A “piecewise function” defines the function in pieces (or parts). In the function below, if x is less than or equal to zero, f(x) = 2x - 1; otherwise, f(x) = x2 - 1.

  8. Domain of a Function Generally, the domain is implied to be the set of all real numbers that yield a real number functional value (in the range). Some restrictions to domain: 1. Denominator cannot equal zero (0). 2. Radicand must be greater than or equal to zero (0). 3. Practical problems may limit domain.

  9. Domain of a Function Generally, the domain is implied to be the set of all real numbers that yield a real number functional value (in the range). Some restrictions to domain: 1. Denominator cannot equal zero (0). 2. Radicand must be greater than or equal to zero (0). 3. Practical problems may limit domain.

  10. Summary of Functional Notation In addition to working problems, you should know and understand the definitions of these words and phrases: dependent variable independent variable domain range function functional notation functional value implied domain

  11. Vertical Line Test for a Function A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

  12. Increasing, Decreasing, and Constant Function On the interval containing x1 < x2, 1. f(x) is increasing if f(x1) < f(x2). Graph of f(x) goes up to the right. 2. f(x) is decreasing if f(x1) > f(x2). Graph of f(x) goes down to the right. On any interval, 3. f(x) is constant if f(x1) = f(x2). Graph of f(x) is horizontal.

  13. Even and Odd Functions 1. A function given by y = f(x) is even if, for each x in the domain, f(-x) = f(x). 2. A function given by y = f(x) is odd if, for each x in the domain, f(-x) = - f(x).

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