1. College Algebra with Modeling and Visualization
Sixth Edition
Chapter 1
Introduction to Functions and Graphs
Copyright © 2018, 2014, 2010 Pearson Education, Inc. All Rights Reserved

2. 1.3 Functions and Their Representations
Learn function notation
Represent a function four different ways
Define a function formally
Use set-builder notation and interval notation
Identify the domain and range of a function
Use calculators to represent functions
Identify functions
Represent functions with diagrams and equations

3. Basic Concepts (1 of 4)
A function is a process or computation that receives inputs and produces outputs. A special characteristic of a function is that for each particular input, it always produces the same output. Although thunder is caused by lightning, we sometimes see a flash of lightning before we hear the thunder. The farther away lightning is, the greater the time lapse between seeing the flash of lightning and hearing the thunder.

4. Basic Concepts (2 of 4)
The following table lists the approximate distance y in miles between a person and a bolt of lightning when there is a time lapse of x seconds between seeing the lightning and hearing the thunder.
x (seconds) 5 10 15 20 25
y (miles) 1 2 3 4
The value of y can be found by dividing the corresponding value of x by 5.

5. Basic Concepts (3 of 4)
This table establishes a special type of relation between x and y, called a function.
x (seconds) 5 10 15 20 25
y (miles) 1 2 3 4
Each x determines exactly one y, so we say that the table represents or defines a function ƒ.
Function ƒ computes the distance y between an observer and a lightning bolt, given the time lapse x. We say that y is a function of x.

6. Basic Concepts (4 of 4) f (Input) = Output
This computation is denoted y = ƒ(x), which is called function notation and is read “y equals ƒ of x.” It means that function ƒ with input x produces output y. That is,
f (Input) = Output

7. Function Notation (1 of 2)
The variable y is called the dependent variable, and the variable x is called the independent variable. The expression f(20) = 4 is read “f of 20 equals 4” and indicates that f outputs 4 when the input is 20. A function computes exactly one output for each valid input. The letters f, g, and h are often used to denote names of functions.

8. Domain and Range of a Function
The set of all meaningful inputs x is called the domain of the function. The set of corresponding outputs y is called the range of the function. A function f that computes the height after x seconds of a ball thrown into the air, has a domain that might include all the times while the ball is in flight, and the range would include all heights attained by the ball.

9. Function Notation (2 of 2)

10. Representation of Functions
Functions can be represented by Verbal Representations (Words) Numerical Representation (Table of Values) Symbolic Representation (Formula) Graphical Representation (Graph)

11. Verbal Representation (Words)
In the lightning example, “Divide x seconds by 5 to obtain y miles.” OR “Function f calculates the number of miles y from a lightning bolt when the delay between thunder and lightning is x seconds.”

12. Numerical Representation (Table of Values)
Here is a table of the lightning example using different input-output pairs (the same relationship still exists):
x (seconds) 1 2 3 4 5 6 7
y (miles)
0.2
0.4
0.6
0.8
1.2
1.4
Since it is inconvenient or impossible to list all possible inputs x, we refer to this type of table as a partial numerical representation.

13. Symbolic Representation (Formula)
In the lightning example,
where y = ƒ(x). We say that function ƒ is represented by, defined by, or given by

14. Graphical Representation (Graph) (1 of 2)
A graph visually pairs and x-input with a y-output. Using the lightning data:

15. Graphical Representation (Graph) (2 of 2)
The scatterplot suggests a line for the graph of f.

16. Four Representations of a Function

17. Formal Definition of a Function
A function is a relation in which each element of the domain corresponds to exactly one element in the range. The ordered pairs for a function can be either finite or infinite.

18. Example: Finding the domain and range
Let a function ƒ be defined by ƒ(−1) = 4, ƒ(0) = 3, ƒ(1) = 4, and ƒ(2) = −2. Write ƒ as a set of ordered pairs. Give the domain and range.
Solution
Because ƒ(−1) = 4, the ordered pair (−1, 4) is in the set. It follows that ƒ = {(−1, 4), (0, 3), (1, 4), (2, −2)}.
The domain D of ƒ is the set of x-values, and the range R of ƒ is the set of y-values.
Thus
D = {−1, 0, 1, 2} and R = {−2, 3, 4}.

19. Set-Builder Notation

20. Example: Evaluating a function and determining its domain (1 of 3)
Let a function f be represented symbolically by
a. If possible, evaluate f(2), f(1), and f(a + 1). b. Find the domain of f. Use set-builder notation. Solution a.

21. Example: Evaluating a function and determining its domain (2 of 3)
Let a function f be represented symbolically by

22. Example: Evaluating a function and determining its domain (3 of 3)
Thus we exclude 1 from the domain (x ≠ 1).

23. Example: Evaluating a function symbolically and graphically (1 of 4)
a. Find the domain and range of g. Use interval notation. b. Use g(x) to evaluate g(−1). c. Use the graph of g to evaluate g(−1).

24. Example: Evaluating a function symbolically and graphically (2 of 4)
Solution a. The domain for g(x) = x² − 2x, is all real numbers or (−∞, ∞).
The minimum y-value on the graph is −1. The arrows on the graph point upward, so there is no maximum y-value. The range is all real numbers greater than or equal to −1, or (−1, ∞].

25. Example: Evaluating a function symbolically and graphically (3 of 4)
Solution b. g(−1) = (−1)² − 2(−1) = = 3 c. Find x = −1 on the x-axis. Move upward to the graph of g. Move across (to the right) to the y-axis. Read the y- value: g(−1) = 3.

26. Example: Evaluating a function symbolically and graphically (4 of 4)

27. Example: Find the domain and range graphically (1 of 2)
a. Evaluate f(1) b. Find the domain and range of f. Use both set builder and interval notation.

28. Example: Find the domain and range graphically (2 of 2)
Solution a. Start by finding 1 on the x-axis. Move up and down on the grid. Note that we do not intersect the graph of f. Thus f(1) is undefined.
b. Arrow indicates x and y increase without reaching a maximum.

29. Graphing Calculators and Functions (1 of 4)
Graphing calculators can be used to create graphs and tables of a function—usually more efficiently and reliably than pencil-and-paper techniques. However, a graphing calculator uses the same basic method that we might use to draw a graph. For example, one way to sketch a graph of y = x² is to first make a table of values, such as x −3 −2 −1 1 2 3 y 9 4

30. Graphing Calculators and Functions (2 of 4)
We can plot these points in the xy-plane, as shown.

31. Graphing Calculators and Functions (3 of 4)
Next we might connect the points with a smooth curve, as shown.

32. Graphing Calculators and Functions (4 of 4)
A graphing calculator typically plots numerous points and connects them to make a graph. In the figure, a graphing calculator has been used to graph y = x².

33. Identifying Functions: Vertical Line Test
If every vertical line intersects a graph at no more than one point, then the graph represents a function. Note: If a vertical line intersects a graph more than once, then the graph does not represent a function.

34. Example: Identifying a function graphically
Use the vertical line test to determine if the graph represents a function.

35. Functions Represented by Diagrams and Equations
There are two other ways that we can represent, or define, a function:
Diagram
Equation

36. Diagrammatic Representation (Diagram) (1 of 2)
Function
Sometimes referred to as mapping: If f(5) = 1 then 1 is the image of 5, and 5 is the preimage of 1.

37. Diagrammatic Representation (Diagram) (2 of 2)
Not a function. Input 2 has two outputs: 5 and 6.

38. Functions Defined by Equations
The equation x + y = 1 defines the function f given by f(x) = 1 − x where y = f(x). Notice that for each input x, there is exactly one output y determined by y = 1 − x.

39. Example: Identifying a function (1 of 2)
Determine if y is a function of x. a. x = y² b. y = x² − 2
Solution a. If we let x = 4, then y could be either 2 or −2. So, y is not a function of x. The graph shows it fails the vertical line test.

40. Example: Identifying a function (2 of 2)
b. y = x² − 2 Each x-value determines exactly one y-value, so y is a function of x. The graph shows it passes the vertical line test.