1 / 23

0.1 Functions and Their Graphs

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

yule
Download Presentation

0.1 Functions and Their Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 0.1 Functions and Their Graphs

  2. 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 • A rational number is a decimal number that may can be written as a finite or infinite repeating decimal, such as or = 2.333…

  3. 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

  4. Real Numbers (3) • We use inequalities to compare real numbers x is less than y x is less than or equal to y x is greater than y x is greater than or equal to y

  5. Real Numbers (4) • We often use a double inequality such as as shorthand for a pair of inequalities andWhen 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.

  6. Real Numbers (5) • Inequalities can be expressed geometrically or by using interval notation.

  7. Real Numbers (6)

  8. Real Numbers (7) • The symbolsanddo not represent actual numbers, but indicate that the corresponding line segments extend infinitely far to the left or right.

  9. 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.

  10. Functions (2) • Examples of functions: f(x) = 3x –1 f(x) = 9x3 + 7x2 – 8

  11. 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

  12. Functions (4) • Function Example: Let f be the function with the domain of all real numbers x, and defined by the formulaf(x) = (4 – x)/(x2 + 7). Find f(h): Find f(h+2): f(h+2) =

  13. 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 is all real numbers greater than or equal to 0.

  14. Graphs of Functions • We can express a function geometrically by expressing it as a graph in a rectangular xy-coordinate 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 xy-plane 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.

  15. Graphs of Functions (2)

  16. Graphs of Functions (3)

  17. 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.

  18. Graphs of Functions (5)

  19. 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?

  20. 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.

  21. Graphs of Functions (7)

  22. 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.

  23. Graphs of Equations (2)

More Related