Increasing & Decreasing Functions A function f is increasing on an interval if, for any x 1 and x 2, in the interval, x 1 < x 2 implies f(x 1 ) < f(x 2 ) A function f is decreasing on an interval if, for any x 1 and x 2, in the interval x 1 f(x 2 ) A function f is constant on an interval if, for any x 1 and x 2, in the interval f(x 1 ) = f(x 2 ). Relative Minimum and Maximum Values A function values f(a) is called a relative minimum of f if there exists an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f(a) ≤ f(x). A function value f(a) is called a relative minimum of f if there exists an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f(a) ≥ f(x).

Graphing Step Functions and Piecewise-Defined Functions Greatest Integer Function- f(x) = the greatest integer less than or equal to x. The graph looks like “steps”: A piece-wise function is a function that is defined by two or more equations over a specified domain. Graphing a piece-wise function: Sketch a graph of each equation on the appropriate portion of the domain. Ex 1:

Even and Odd Functions A graph is symmetric with respect to the y-axis if (x, y) as well as (-x, y) is on the graph. A graph is symmetric with respect to the origin if (x, y) as well as (-x, -y) is on the graph. A graph is symmetric with respect to the x-axis if (x, y) as well as (x, -y) is on the graph. A function f is EVEN if, for each x in the domain of f, f(-x) = f(x). A function f is ODD if, for each x in the domain of f, f(-x) = - f(x). Ex 1: Is the function f(x) = |x| even, odd, or neither? Algebraic way?Graphical way?

Ex 2: g(x) = x 3 - x Algebraic way?Graphical way? Ex 3: h(x) = x Algebraic way?Graphical way? Ex 4: f(x) = x 3 -1 Algebraic way?Graphical way?