Function and their graphs, Summaries of Mathematics

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GENERAL MATHEMATICS

PERFORMANCE TASK

FUNCTION AND GRAPHS

• FUNCTION FUNCTIONS A function from a set X to a set Y is a rule or correspondence that associates with each element of X exactly one element of Y. (In Math, these sets usually consist of real numbers.) With the function notation y=f(x), each x value has only one corresponding value. You can think of a function as being like a program. The x-values are the inputs, and the y-values are the outputs. The possible inputs form the domain of the function, and the possible outputs form its range. For the functions that we are dealing with, the numbers that we need to exclude from the domain are numbers that lead to division by zero, or the square root of a negative number. (If the function was a program, trying to divide by zero or take the square root of a negative number would give an error message.) Given two functions, we can make a new function from their sum, difference, product, or quotient.

PROPERTIES OF FUNCTION A function is called even if f(−x) = f(x) (the graph is symmetric about the y-axis) and odd if f(−x) = −f(x) (the graph that is symmetric about the origin). A function is increasing on an interval if its values keep going up, and decreasing on an interval if its values keep going down. A high point on the graph is called a local.

FUNCTION AND THEIR GRAPHS maximum, and this corresponds to a change from increasing to decreasing. A low point on the graph is called a local minimum, and corresponds to a change from decreasing to increasing. Note: Since we are not using calculators, you won’t be asked to actually compute local maximum and local minimum values in this section. The average rate of change of a function is found by dividing the change in y by the change in x. If you go from x to con the x-axis, then the corresponding change in yis f(x)−f(c). We get this formula for the average rate of change: f(x)−f(c) x− c, where x6=c.

GRAPHING TECHNIQUES:TRANSPORMATION The basic model for a linear function is f(x) = x, whose graph is a straight line through the origin that slopes up at a 45oangle. The family of linear functions includes all functions of the form f(x) = ax +b. We can get all of these by multiplying the basic example by and adding b. The numbers and tell us all about the new line: if is positive it slopes up, if is negative it slopes down; gives the y-intercept, and tells how far the line has been moved up (if b > 0) or down (if b < 0).