Transcript show1
0.1 Functions and Their Graphs Real Numbers • A set is a collection of objects. • The real numbers represent the set of numbers that can be represented as decimals. • We distinguish two different types of decimal numbers o A rational number is a decimal number that may can be written as a finite or infinite repeating decimal, such as 2 or = 2.333… Real Numbers (2) An irrational number is a decimal number that does not repeat and does not terminate, such as = -1.414214… and = 3.14159… • The real number set is represented geometrically by the real number line Real Numbers (3) • We use inequalities to compare real numbers x y x is less than y x y x is less than or equal to y x is greater than y xy x is greater than or equal to y x y Real Numbers (4) • We often use a double inequality such as abc as shorthand for a pair of inequalities ab and bc When we use double inequalities, the positions of a, b, and c must be written as they would appear on the real number line if read from right to left or left to right. Real Numbers (5) • Inequalities can be expressed geometrically or by using interval notation. Real Numbers (6) Real Numbers (7) • The symbols and do not represent actual numbers, but indicate that the corresponding line segments extend infinitely far to the left or right. Functions • A function of a variable x is a rule f that assigns to each value of x a unique number f(x) (read ”f of x”), called the value of the function at x. • x is called the independent variable. • The set of values that the independent variable is allowed to assume is called the domain of the function. A function’s domain might be explicitly specified as part of its definition or might be understood from its context. • The range of a function is the set of values that the function assumes. Functions (2) • Examples of functions: f(x) = 3x –1 f(x) = 9x3 + 7x2 – 8 Functions (3) • Function Example: If we let f be the function with the domain of all real numbers x, and defined by the formula f(x) = 4x2 – 2x + 6, we can find the corresponding value in the range of f for a given value of x by substitution. Find f(2): f(2) = 4(2)2 – 2(2) + 6 = 4(4) – 4 + 6 = 16 – 4 + 6 = 18 Find f(-3): f(-3) = 4(-3)2 – 2(-3) + 6 = 4(9) + 6 + 6 = 36 + 6 + 6 = 48 Functions (4) • Function Example: Let f be the function with the domain of all real numbers x, and defined by the formula f(x) = (4 – x)/(x2 + 7). 4h Find f(h): f (h) 2 h 7 Find f(h+2): f(h+2) = 4 (h 2) 2 (h 2) 7 Functions (5) • When the domain of a function is not explicitly specified, it is understood that the domain of the function is all values for which the defining formula makes sense. The domain of f for f(x) = 2x is all real numbers. The domain of f for f(x) = 1/x is all real numbers except x = 0. The domain of f for f ( x) x is all real numbers greater than or equal to 0. Graphs of Functions • We can express a function geometrically by expressing it as a graph in a rectangular xycoordinate system. • Given any x in the domain of a function f, we can plot the point (x, f(x)). This is the point in the xyplane whose y-coordinate is the value of the function at x. • It is possible to approximate the graph of the function f(x) by plotting the points (x,f(x)) for a representative set of values of x and joining them by a smooth curve. • We often use a tabular construct to capture the points we wish to plot. Graphs of Functions (2) Graphs of Functions (3) Graphs of Functions (4) • To every x in its domain, a function assigns one and only one value of y. That value is precisely f(x). This means: The variable y is called the dependent variable because its value depends on the value of the independent variable x. – For every curve that is the graph of a function, there is a unique y such that (x,y) is a point on the curve. Not every geometric curve is the graph of a function. Graphs of Functions (5) Graphs of Functions (6) • Vertical Line Test – A curve in the xy-plane is the graph of a function if and only if no vertical line can be drawn that will intersect the curve at more than one point. • Which curves are the graphs of functions? Three Views of a Function • We how have three ways to describe a function • By giving the formula f(x) = … and defining the domain of the independent variable (x). A function specified in terms of a formula is said to be defined analytically. • By drawing the functions graph. Such a function is said to be defined graphically. • By providing a table of function values (x and f(x)). This method is said to describe a function numerically. Graphs of Functions (7) Graphs of Equations • The equations arising in connection with functions all have the form: y = [an expression in x] • Note that not all equations connecting x and y are functions. • A graph of an equation can be plotted the same as the graph of a function. However, the resulting graph will only pass the Vertical Line Test if the equation represents a function. Graphs of Equations (2)