Section 1.2

1. Section 1.2 Functions and Graphs

2. Relation A relation is a correspondence between the first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to at least one member of the range. Domain: set of all values of x Range: set of all values of y

3. Function A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range. Function Not a Function ?7 49 ?2 4 7 5 7 0 0 6 3 ?2 4 8 2

4. Function? Which of the following relations is a function? {(8, ?2), (8, 4), (7, 3)} {(?6, 4), (1, 4), (7, 4)}

5. Notation for Functions The inputs (members of the domain) are values of x substituted into the equation. The outputs (members of the range) are the resulting values of y. f(x) is read “f of x,” or “f at x,” or “the value of f at x.” Example: Given f (x) = 3x2 ? 4, find f (6).

6. Function We are going to be dealing with functions where we think of the . . . function- as a rule that tells how to determine the dependent variable (y) for a specific value of the independent variable (x).

7. Four Ways to Represent a Function Verbally (description in words) Numerically (by a table of values) Visually (by a graph) Algebraically (by a formula)

8. A Function Verbally ex. The amount of sales tax depends on the amount of the purchase.

9. A Function Numerically Use a table of values: x y 19 0 10 -1 7

10. A Function Visually Any function can be visually represented by a graph.

11. A Function Algebraically f(x) = 3x + 10 To solve a function: Evaluate the function.

12. Function Notation “y is a function of x” y depends on x y = f (x) f (x) is just another name for y ! ex. f(x) = 3x + 10

13. Vertical Line Test If every vertical line intersects a graph in no more than one point, then the graph is the graph of a function.

14. Example: Does the graph represent a function? The graph is a function because we cannot find a vertical line that crosses the graph more than once.

15. Example: Does the graph represent a function? The graph is not a function. We can find a vertical line that crosses the graph in more than one point.

16. Finding Domains of Functions Find the indicated function values and determine whether the given values are in the domain of the function. f (1) and f (5), for f(1) = Since f(1) is defined, 1 is in the domain of f. f(5) = Since division by 0 is not defined, the number 5 is not in the domain of f.

17. Another Example Find the domain of the function Solution: We can substitute any real number in the numerator, but we must avoid inputs that make the denominator 0. Solve x2 ? 3x ? 28 = 0. (x ? 7)(x + 4) = 0 x ? 7 = 0 or x + 4 = 0 x = 7 or x = ?4 The domain consists of the set of all real numbers except ?4 and 7 or {x|x ? ?4 and x ? 7}.

18. Visualizing Domain and Range Keep the following in mind regarding the graph of a function: Domain = the set of a function’s inputs, found on the horizontal axis. Range = the set of a function’s outputs, found on the vertical axis.

19. Domain and Range(How to find domain and range graphically) Domain The set of a function’s inputs, x-values, found on the horizontal axis Range The set of a function’s outputs, y-values, found on the vertical axis

20. Example Graph the function. Then estimate the domain and range.

21. When to restrict Domain?(How to find domain algebraically) Fractions You can’t have zero in the denominator! Even root radical You can’t have a negative under an even root radical.