1.1 - Functions

Interval Notation x > 3 -1 ≤ x < 5 x < -2 or x ≥ 4 Algebraic Notation Graph Interval Notation x > 3 -1 ≤ x < 5 x < -2 or x ≥ 4 | 3 | -1 | 5 | -2 | 4

Function, Domain, and Range A function is a relationship or correspondence between two sets of numbers, in which each member of the first set (called the domain) corresponds to one an only one member of the second set (called the range).

f x1 y1 x2 y2 x3 y3 X Y Range Domain

Functions? State the domain and range. -4 1 6 -8 -2 5 7 -2 3 5 1 7

Explanations If asked, how do you state that a relation is or is not a function? Yes, it is a function. Each domain element corresponds to exactly one range element. No, it is not a function. Some domain element corresponds to more than one range value.

Function Representation Verbal or Written Numerically Graphically Symbolically

Examples 1. A correspondence between the students in this class and their student identification numbers. Do the following relations represent functions?

y Function? x

y x

Theorem Vertical Line Test If any vertical line drawn on the graph of a relation crosses the graph more than once, the relation does not represent a function. Caution: It is not sufficient to state that a graph represents a function because it passes the vertical line test.

Determine the domain, range, and intercepts of the following graph. y 4 (2, 3) (10, 0) (4, 0) (1, 0) x (0, -3) -4

Example Does this equation represent a function? Why or why not? (x – 2)2 + (y + 4)2 = 25

Linear Functions y = mx + b [Slope-Intercept Form of a Line] m = slope b = y-intercept y – y1 = m(x – x1) – [Point-Slope Form of a Line] m = slope (x1, y1) = point on line x = x1 [Vertical Line] m = undefined or none y = y1 [Horizontal Line] m = 0

Linear Functions - Examples Determine the equation of the line through (-2, ¾) with slope m = ½ . Determine the equation of the line through (4, -1) with no slope. Determine the equation of the line through (-2, 5) with m = 0.

Example Let y = 2x – 3 y = x2 + 2x – 1 y = | x + 5 | If x = 3, then y = _____? There is only one answer to this question.

Example Let f = 2x – 3 g = x2 + 2x – 1 h = | x + 5 | If x = 3, then h = _____? There is only one answer to this question.

f (x) Function Notation Does Not Mean f times x. Function Notation f (x) Read: “f of x” f is the name of the function x is the variable into which we substitute values or other expressions x is called the independent variable and f(x) is the dependent variable.

Find the domain of the following functions: (b) (c)

Evaluating Functions Let f(x)=2x – 3 and g(m) = | m2 – 2m + 1|. Determine: f(-3) g(-2) f(a) g(a + 1)

The Difference Quotient Example If f (x) = x2 – 3, determine the difference quotient.

The Difference Quotient First, we need to determine f (x + h) f (x + h) = ( )2 – 3 x + h x = (x2 + 2hx + h2) – 3 = x2 + 2hx + h2 – 3 f (x) = x2 – 3

The Difference Quotient Example Determine the difference quotient for each of the following and simplify.

Increasing and Decreasing Functions A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2 we have f(x1) < f(x2). A function f is decreasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2 we have f(x1) > f(x2). A function f is constant on an open interval I if, for all choices of x, the values f(x) are equal.

Increasing, Decreasing Constant, Local Maximum, Local Minimum y 4 (2, 3) (4, 0) (1, 0) x (10, -3) (0, -3) (7, -3) -4

Piece Wise Defined Functions When functions are defined by more than one equation, they are called piece-wise defined functions.

Example For the following piece-wise defined function: Find f (-1), f (1), f (3). Sketch a graph of f. Find the domain of f.

Absolute Value

Example Use the definition to do the following: Explain why | -2 | = 2. Determine the exact value of | 3 – π |. Write | x – 2 | as a piece wise defined function without absolute value bars.

Modeling With Functions Example 1 Express the surface area of a cube as a function of its volume. Example 2 A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window.

Odd or Even Function A function is odd if f (-x) = -f (x). Odd functions exhibit origin symmetry on their graphs. This means if you turn the graph upside down, it will look the same. A function is even if f (-x) = f (x). The graphs of even functions will be symmetric to the y-axis. Simply substitute -x for each x in the function and determine if you get f (x) or -f (x). If you get neither, it is neither odd nor even.

Odd or Even Function Determine if is odd or even.